2013 Tiling systems and 2-graphs associated to textile systems

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Tiling systems and 2-graphs associated to textilesystemsYuxiang TangUniversity of Wollongong

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Recommended CitationTang, Yuxiang, Tiling systems and 2-graphs associated to textile systems, Doctor of Philosophy thesis, School of Mathematics andApplied Statistics, University of Wollongong, 2013. http://ro.uow.edu.au/theses/4003

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Tiling systems and 2-graphs associated totextile systems

A thesis submitted in fulfilment of the requirements for the award of the degree

Doctor of Philosophy in Mathematics

from

University of Wollongong

by

Yuxiang Tang

AUGUST, 2013

CERTIFICATION

I, Yuxiang Tang, declare that this thesis, submitted in partial fulfilment of the

requirements for the award of Doctor of Philosophy, in the School of Mathematics and

Applied Statistics, University of Wollongong, is wholly my own work unless otherwise

referenced or acknowledged. The document has not been submitted for qualifications

at any other academic institution.

Your name here

ACKNOWLEDGEMENTS

Undertaking this PhD has been a truly life-changing experience for me and it

would not have been possible without the support and guidance that I received from

many people.

First of all, I want to honor almighty God who is so great and full of compassion

and who has provided me the opportunity to proceed successfully.

The thesis appears in its current form due to the assistance and guidance of

several people. I would like to offer my sincere appreciation to all of them. I would

like to thank my esteemed PhD advisor, Professor Aidan Sims, for supporting me

during these past three and half years. I would like to express my cordial thanks

for grooming me as a PhD student, your warm encouragement, thoughtful guidance,

insightful decisions, critical comments and correction of the thesis. I hope that I

can be as intelligent, enthusiastic, and diligent as Aidan and to someday be able to

undertake research work as he does. Without his guidance and constant feedback this

PhD would not have been achievable.

I also have great appreciation to Associate Professor David Pask who has been so

supportive and has given me the freedom to pursue various projects without objection.

He has also provided insightful discussions about the research. I am also very grateful

to David for his scientific advice and knowledge and many insightful discussions and

suggestions. Although David’s work load has been heavy which includes supervising

other students, as well as doing his own research, he has helped significantly in giving

me a clear vision for my project. David is clear and patient with any explanation he

gives.

I greatly appreciate the Operator Algebra Seminar Program which has expanded

my insight and provided me inspiration for my projects. I would thank Sam Webster,

Michael Whittaker, Nathan Brownlowe and Hui Li who have also given me valuable

iv

advice and encouragement during my PHD research.

My special thank goes to my family in China for their love and support, and to

my wife for her patience and understanding during the period of my study. Also,

this thesis is for my son James Tang. Finally, all great thanks and praise to God

for allowing me to complete a successful thesis, meeting wonderful magnificent people

here at University of Wollongong, and guiding me and giving me the best opportunity

of my lifetime.

Abstract

This thesis concerns dynamical systems called tiling systems, and combinatorial

structures called 2-graphs associated to textile systems.

Textile systems were developed as a means of defining and studying two dimen-

sional dynamical systems called shift spaces. We show how to construct a 2-graph

from a textile system when the system has what we call unique path lifting, and

investigate the relationship between the properties of the 2-graph and the dynamical

properties of the tiling system.

Our motivation is to link the properties of a 2-graph which appear in hypotheses

of theorems about C∗-algebras to key properties of dynamical systems.

Specifically we have investigated the textile systems for higher block codings of

tiling systems. We show that the 2-dimensional dynamical system associated to the

2-graph constructed from a textile system is conjugate to the tiling systems corre-

sponding to the textile system. We investigate the periodicity of tilings arising from

textile systems and give sufficient conditions to guarantee that there exist some tilings

which are periodic. We also use ideas developed for 2-graphs to provide checkable

conditions on a textile system under which the associated shift space is topologically

transitive.

We study the relationship between topological properties of the shift space of a

textile system and the connectivity of the associated 2-graph. We also investigate the

entropy of the tiling system, and show that it is zero when the textile system gives

rise to a 2-graph. We construct examples of textile systems whose associated shift

spaces have arbitrarily small nonzero entropy.

CONTENTS

Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Notations and basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Textile systems and 2-graphs . . . . . . . . . . . . . . . . . . . . . . . . . 27

4. Textile systems and tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5. Aperiodicity of shift spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1 Aperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Beatty Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

1. INTRODUCTION

1.1 Background

1.1.1 Literature review

Symbolic dynamics originated in 1898 from Hadamard’s study [16] of the geodesic

flows on surfaces of negative curvature. Symbolic dynamics provides a good machine

to study dynamical systems. It has wide applications in studying data storage and

transmission, and it is also a useful tool to understand information systems or me-

chanical systems. It has been applied to model motion of molecules in a gas and

many other related areas.

In 1921, Morse used this idea [37], [38] to study the construction of a nonperiodic

recurrent geodesic; later, Artin [3] in 1924 did related work on the system which is

now called Artin billiards. Morse and Hedlund [39] formalized the subject of symbolic

dynamics 75 years ago. Later discoveries of Morse and Hedlund developed this theory

in their field [40]. By applying symbolic sequences and shifts of finite type, in 1948,

Shannon [60] developed information theory.

In symbolic dynamics, a shift space or subshift is a set of infinite words that

represent a discrete system. Let A be a finite set. Its elements are called symbols,

and the set A is called the alphabet. Symbolic dynamical systems are often defined

by a set of forbidden blocks – that is finite sequences of symbols from the alphabet.

Let F be a collection of blocks over A which we regard as being the forbidden blocks.

The associated shift XF is the collection of infinite words that do not contain any

block occurring in F . A shift space over the alphabet A is a subset X of AZ such

that X = XF for some collection F of forbidden blocks over A.

We use (X, σ) to denote a shift space X to emphasize the role of the shift operator.

1. Introduction 2

The full-A shift is the collection of all bi-infinite sequences of symbols of A. The shift

map σ over the full-shiftAZ maps a point x to the point y = σ(x) whose ith. coordinate

is yi = xi+1. For instance, if x = . . . x−2x−1 · x0x1x2 . . ., then y = x−2x−1x0 · x1x2 . . .,

so that σ(x)i = xi+1 for all i ∈ Z. A point x ∈ A is periodic for σ if σn(x) = x for

some n ≥ 1 and we say that x has period n under σ.

In Lind and Marcus’s book [31], the theory of one-dimensional symbolic dynamics

is explained excellently. However, the dynamics of multi-dimensional shifts is much

more complex than that of one-dimensional shifts. We define σ(1,0) and σ(0,1) in AZ2

by σ(1,0)(x)(m,n) = x(m + 1, n) and σ(0,1)(x)(m,n) = x(m,n + 1). We call these

the horizontal and vertical shift maps on AZ2. A 2-dimensional shift space over the

alphabet A is a closed subset X of AZ2which is invariant under both vertical and

horizontal shifts. More generally, for (m,n) ∈ Z2, we define the shift map σ(m,n) by

σ(m,n)(x)(i, j) = x(i + m, j + n).

In this thesis, we mainly study shift spaces associated with textile systems. Tex-

tile systems introduced by Nasu [43] are useful and powerful machines to study the

dynamical behavior of endomorphisms and automorphisms of topological Markov

shifts. Nasu has given many interesting results regarding textile systems (see for ex-

ample [42] [43], and [44]). A textile system (E, F, p, q) is defined by directed graphs

F and E and an ordered pair of surjective graph morphisms p, q : F → E such that

the map f → (p(f), q(f), r(f), s(f)) is injective on F 1; that is (p(f), q(f), r(f), s(f))

completely determines f .

A textile system gives rise to a system of Wang tiles which may then be used to

build a shift space. We use Wang tiles to construct shift spaces later. The idea of

Wang tiles was originally introduced by Wang [17] in 1961. Wang tiles are square tiles

with colored edges. The tile t whose west, east, north and south edges are coloured

by w(t), e(t), n(t) and s(t) is described as the 4-tuple (w(t), e(t), n(t), s(t)). A tile set

is a finite set of Wang tiles.

A tiling of the infinite Euclidean plane by a set of Wang tiles T is an arrangement

of copies of the tiles in the tile set such that the tiles cover the whole plane and

adjoining edges always have the same colour. Formally, we encode this as a function

x : Z2 → T such that n(x(p, q)) = s(x(p, q + 1)), and similarly for the other three

1. Introduction 3

sides. Let (T, K, n, e, s, w) be a system of Wang tiles, where K is the colour set. Let

x : Z2 → T be a tiling. Then σ(m,n)(x) is also a tiling. The tiling x is said to be

periodic with period (m,n) ∈ Z2\(0, 0) if and only if σ(m,n)(x) = x. A tile set T

is called aperiodic if and only if (i) there exists a valid tiling, and (ii) there does not

exist any periodic valid tiling.

Wang conjectured that there exists no aperiodic tile set T . In 1966 [5], Berger

proved that Wang’s conjecture is false by constructing an aperiodic set containing

20426 tiles. Later, Berger reduced the number of tiles to 104. Since then, a number of

researchers have tried to find the smallest cardinality of an aperiodic set of Wang tiles.

In 1966, smaller aperiodic sets were found by Knuth [22] (92 tiles); in 1966, Lauchli

[15] (40 tiles); In 1973, Robinson [15] (32 tiles); and in 1978, Ammann [57](16 tiles).

By using sequential machines that multiply real numbers by rational constants, Kari

developed a new method of constructing aperiodic sets in [20]. Using this approach,

a new aperiodic set containing 14 tiles over 6 colors was found. In 1996, Kari [11]

described an aperiodic set consisting of 13 tiles over 5 colors.

Graphs of rank k, or k-graphs, were introduced by Kumjian and Pask in 2000

[26]. They can be thought of as the path categories of higher-dimensional analogues

of directed graphs, the difference being that paths in k-graphs have a degree or shape

in Nk whereas paths in a directed graph have length in N. In [18], a k-coloured graph

is defined as a directed graph together with a colour map c assigning a colour in

1, . . . , k to each edge. In [18], the authors constructed a directed graph, called a

skeleton EΛ from a k-graph Λ, and described the data needed to recover Λ from EΛ.

The k-graph Λ is row-finite and has no sources if each vertex receives at least one

and at most finitely many paths of each possible degree. Let Λ be a row-finite k-graph

with no sources. In [55], the authors identified notions of aperiodicity and cofinality

for k-graphs; these definitions were motivated by the study of associated C∗-algebras,

and have profound implications for their structure: Λ is cofinal and aperiodic if and

only if its C∗-algebra is simple. It is important to note that aperiodicity has a different

meaning in the context of k-graphs than of tilings – a tile set is aperiodic if it admits no

periodic tilings, but for a k-graph to be aperiodic, we require only that the aperiodic

infinite paths in the k-graph are dense in the set of all infinite paths (see Chapter 3

1. Introduction 4

for detail). Robertson and Sims proved that a k-graph is aperiodic if and only if no

periodicity at v for every v ∈ Λ.

Higher-rank graphs arose out of the work of Robertson and Steger. Let S be a

finite set. In [56], Robertson and Steger considered collections of S×S 0, 1-matrices

M satisfying a list of technical conditions (H0)–(H3).

Any collection of matrices satisfying (H0) and (H1) determine a k-graph [2];

conditions (H2) and (H3) guarantee that the k-graph is cofinal and aperiodic. Every

2-graph determines a textile system and hence a set of Wang tiles; but not every

textile system comes from a 2-graph. However, versions of Robertson and Steger’s

conditions (H2) and (H3) make sense for textile systems, and in [2], a checkable

aperiodicity condition for k-graphs based on (H2) and (H3) is given. An interesting

question explored in this thesis (see § 5.1) is whether these conditions are sufficient

to guarantee aperiodicity and irreducibility of the shift spaces associated to textile

systems.

In [31], a one-dimensional shift space X is called irreducible if whenever u and w

are allowed blocks, there is a “connecting block” v such that uvw is allowed in X. In

higher-dimensional shift spaces, irreducibility is more complicated. Quas and Trow in

[52] gave the definition of irreducibility of higher-dimensional shift spaces. To study

the aperiodicity of shift spaces associated to Wang tile systems (T, K, n, e, s, w), we

draw on intuition coming from the situation (T, K, n, e, s, w) arises from a 2-graph.

Another field we are interested in is topological entropy. The notion of topological

entropy was suggested to measure the complexity of a function space. Roughly, it

measures the exponential growth rate of the number of distinguishable orbits as time

advances. The original definition of entropy was introduced by Adler, Konheim and

McAndrew [1] in 1965. Their idea of assigning a number to an open cover to measure

its size was inspired by Kolmorgorov and Tihomirov [23].

The topological entropy of a continuous map describes roughly speaking the

largest growth rate of the number of covers of the map. In most cases, the map

in question is a shift map and the entropy is positive. It is well known that a topo-

logically mixing shift of finite type (SFT) in one dimension has positive topological

1. Introduction 5

entropy and a unique measure of maximal entropy. In final chapter, we investigate

topological entropy of shift spaces arising from the textile systems.

1.1.2 Main results

In chapter 3, we study textile system (E, F, p, q) and the associated 2-graph

Λ(E, F, p, q) when p, q have unique path lifting. We introduce definitions of [unique]

path lifting, cofinality of 2-graphs, aperiodicity of 2-graphs and dual graphs. We give

some interesting results as follows:

(1). In Theorem 3.8, we find that given a textile systems (E, F, p, q) with unique

path lifting, there exists a unique 2-graphs Λ(E, F, p, q) such that Λe1 = E1, Λe2 = F 0.

Every element of Λe1+e2 is of the form ϕf for some f ∈ F 1. In particular, q(f)s(f) =

r(f)p(f) in Λ for all f ∈ F 1. Moreover, in Theorem 3.20, we prove that the shift

space XT of this textile system is conjugate to the topological space Λ∆ of bi-infinite

paths in Λ.

We then study cofinality of 2-graphs.

(2). Given a textile system (E, F, p, q) with unique path lifting, we prove in

Lemma 3.22 that if E is cofinal, then the associated 2-graph Λ(E, F, p, q) is also

cofinal. We also show in Example 3.24 which illustrates that the converse of this

result fails.

(3). Suppose that (E, F, p, q) is a textile system with unique path lifting. In

Lemma 3.29, we give necessary and sufficient conditions for the associated 2-graph Λ

to be strongly connected. When E0 is finite, Corollary 3.30 simplifies these conditions

substantially.

(4). We then study the aperiodicity of the 2-graph arising from a textile system

(E, F, p, q). In Lemma 3.35, we show that if Λ is aperiodic, then E is aperiodic as

well. However, Example 3.41 shows that the converse of this conclusion fails.

We also describe the dual system (G,H, π, φ) associated to a textile system

(E, F, p, q) (see [13]), and use it to study the aperiodicity of the associated 2-graph.

In Proposition 3.40, we get a symmetric version of the result of (4).

1. Introduction 6

(5). At the end of this chapter, we study the relationship between dual graphs

(as introduced in [2]) and textile systems. Specifically, if (E, F, p, q) is a textile

system, then passing to dual graphs in a suitable way gives a new textile system

(E(m,m + n), F (m,m + n), p, q), and in Theorem 3.48, we show that the resulting

2-graph is isomorphic to a suitable 2-graph dual of Λ.

In Chapter 4, we study textile systems (E, F, p, q) and the tilings arising from the

associated Wang tiles system (T, K, n, e, s, w). Based on the idea of textile system and

essential graphs, we investigate textile system (Eess, Fess, p, q) arising from (E, F, p, q)

and its tilings. As a source of examples of textile systems, we consider skew-product

graphs [26] and their associated textile systems (E×cG,E, π, απ). At the end of this

chapter, we discuss the periodicity of shift spaces and find some sufficient conditions

to guarantee that a textile system admits a periodic tiling. The following are the

main results of this chapter:

(1). We then study the textile system (E, F, p, q) and its essential graphs. In

Lemma 4.18, we find that: if E, F are locally finite graphs and the textile system

(E, F, p, q) has path lifting, then so does (Eess, Fess, p, q). Moreover, the tilings arising

from (E, F, p, q) are exactly the tilings arising from (Eess, Fess, p, q).

(2). Given a discrete group G, given a directed graph E, and a labeling c from

E1 to G, we consider the skew-product graph E ×c G of [24]. In Lemma 4.23, we

describe the bi-infinite paths of skew-product graphs. In Lemma 4.24, we also build

a textile system with unique path lifting from the skew-product graph by adding

specified graph morphisms.

(3). We then investigate periodicity of tilings arising from textile systems. In

Lemmas 4.27– 4.30, sufficient conditions are given to guarantee that there exist some

tilings which are horizontally or vertically periodic and (a, b)-periodic for general

(a, b) ∈ Z2.

In Chapter 5, we use ideas due to Robertson and Steger [56] to study aperiodicity

of tiling associated to textile systems. A textile system with unique path lifting

(E, F, p, q) determines a 2-graph Λ = Λ(E, F, p, q). The transition matrices for the red

and blue graphs give nonnegative integer matrices M1, M2 which commute. In [56],

1. Introduction 7

Robertson and Steger gave checkable conditions (H0)–(H3) to determine cofinality

and aperiodicity of k-graphs. We find that conditions (H2) and (H3) are also useful

to determine the aperiodicity of textile systems. The main points of this chapter are

as follows:

(1). In Theorem 5.3, we show that given matrices M1 and M2 satisfying conditions

(H0)–(H1), we may form a 2-graph Λ(W,d) from the m-blocks determined by M1 and

M2 with composition defined by (H1).

(2). A nonnegative integer matrix may be used to determine a directed graph. The

matrix is then called the transition matrix of the graph. For a 2-graph Λ determined

by a textile system, if we take the dual graph (1, 1)Λ, then the transition matrices

of the red and blue graphs satisfy (H0). Using the ideas of transitions matrices and

dual graphs, we then check conditions (H0)–(H3) for a series of Examples 5.4–5.8

determined by textile systems, and determine the aperiodicity of these 2-graphs.

(3). Based on the examples we have discussed in (2), in Proposition 5.9, we

give sufficient conditions to guarantee the dual graph (1, 1)Λ are (p, 0)-aperiodic and

(0, p)-aperiodic for all p ∈ Z.

(4). We then study the aperiodicity of shift spaces associated to textile systems

with path lifting, and we give checkable conditions for shift spaces to be aperiodic.

Theorem 5.10 says that when a textile system satisfies these hypotheses, the associ-

ated shift space X of the tiling system is topologically free.

(5). We then study topological transitivity of textile systems with path lifting. In

Lemma 5.12, we give sufficient conditions to guarantee that the shift space associated

to a textile system is topologically transitive.

(6). Next, we study the irreducibility of a shift space. Corollary 5.20 proves that if

a shift space (X, σ) is irreducible, then σ is topologically transitive. However, we give

Example 5.21 and Example 5.22 to show that converse fails: we find 1-dimensional and

2-dimensional shift spaces X which are topologically transitive, but not irreducible.

(7). At the end of Chapter 5, we use Beatty Sequences and Culik’s aperiodic tile

set of 13 tiles to build a textile system in Example 5.23, whose shift space has no

1. Introduction 8

periodic points. This example can not arise from a 2-graph as the textile system does

not have path lifting.

In Chapter 6, we study topological entropy of shift spaces associated to textile

systems. We prove the following:

(1). Consider a textile system (E, F, p, q) such that p, q have path lifting and let

Λ = Λ(E, F, p, q) be the associated 2-graph. In Proposition 6.3, we give a proof based

on textile systems that the entropy of the associated shift space X = (Λ∆, σ) is zero

(see also [30]).

(2). However, not all the entropies of shift spaces arising from arbitrary textile

systems are zero. We give some conditions which force textile systems to have positive

entropy. We also construct a family of textile systems in Example 6.8 whose associated

shift spaces have arbitrarily small positive entropy.

2. NOTATIONS AND BASIC FACTS

In this section, we will introduce some background of shift spaces, path lifting,

textile systems and k-graphs.

Definition 2.1: Let A be a finite set. Given x : Z2 → A, we define σ(1,0) and σ(0,1)

in AZ2by

σ(1,0)(x)(m,n) = x(m + 1, n) and σ(0,1)(x)(m,n) = x(m,n + 1).

We call these the horizontal and vertical shift maps on AZ2. A 2-dimensional shift

space over the alphabet A is a topologically closed subset X of AZ2which is closed

under both vertical and horizontal shifts.

For more details of the topology of shift spaces, see Chapter 3. Many of the shift

spaces discussed in this thesis will be constructed from directed graphs. The following

definition sets out to formalize directed graphs.

Definition 2.2: A directed graph is a quadruple (E0, E1, r, s), where E0, E1 are

countable sets, and r, s : E1 → E0 are functions.

We interpret this visually as a collection E0 of points and a collection E1 of arrows

connecting the points in E0, where the arrow e points from s(e) to r(e). In this thesis,

we assume that every graph is infinite graph unless we state specifically.

There may be more than one edge between two vertices. An edge e with r(e) =

s(e) is called a loop. For v ∈ E0, we write s−1(v) for the set of edges which begin at

v and write r−1(v) for the set of edges which end at v.

Convention 2.3: A path π = e1e2 . . . en in directed graph E is a finite sequence of

edges ei from E1 such that s(ei) = r(ei+1) for 1 ≤ i ≤ n − 1. Our convention for

2. Notations and basic facts 10

paths is consistent with the categorical approach to graphs which will pursue later.

The length of π = e1e2 . . . en is |π| = n, the number of edges it traverses. Define En

be the collection of paths with length of n in E. The path π = e1e2 . . . en starts at

vertex s(π) = s(en) and ends at vertex r(π) = r(e1), and π is a path from s(π) to

r(π). A cycle is a path that starts and ends at the same vertex. A simple cycle is a

cycle that does not intersect itself, i.e., a cycle π = e1e2 . . . en such that the vertices

s(e1), s(e2), . . . , s(en) are distinct. For each vertex v of E0 there is also an empty path

having length 0, which both starts and ends at v. We denote this path v as well.

Definition 2.4: Let E be a directed graph, we write E∗ for the collection of finite

paths in E, so E∗ =⋃

n≥0 En.

Remark 2.5: In [27] and [28](for example), the term “loop” was used to refer to

what we have called a cycle. We have chosen to adopt the graph theory convention

here, where a loop usually refers to a single edge with the same range and source.

Definition 2.6: Let E, F be directed graphs. A graph morphism from F to E con-

sists of a pair of maps φ0 : F 0 → E0 and φ1 : F 1 → E1 such that

s(φ1(f)) = φ0(s(f)) and r(φ1(f)) = φ0(r(f))

for all f ∈ F 1. We write φ = (φ0, φ1) : F → E. If (φ0, φ1) are one-to-one then φ is

called a graph embedding. If they are bijective, then φ is called a graph isomorphism

and we write F ∼= E.

When two graphs are isomorphic we can obtain one from the other by renaming

vertices and edges. Thus, we tend to regard isomorphic graphs as being “the same”.

Definition 2.7: An automorphism of a graph E is a bijective graph morphism α :

E → E.

We now introduce the notion of Wang tiles which will be used later to construct

shift spaces.


Definition 2.8: A system (T, K, n, e, s, w) of Wang tile consists of finite sets T and

K and functions n, e, s, w : T → K. Given a Wang tile system (T, K, n, e, s, w), we

call T the tile set and K the colour set.

We picture the elements t of T in a system (T, K, n, e, s, w) of Wang tiles as square

tiles whose edges are coloured n(t), s(t), e(t), w(t) as follows:

t

Figure 2.1. The configuration of tiles.

Definition 2.9: Let F and E be directed graphs, and let p, q : F → E be sur-

jective graph morphisms. Then (E, F, p, q) is called a textile system if the map

f → (p(f), q(f), r(f), s(f)) is injective on F 1; that is (p(f), q(f), r(f), s(f)) com-

pletely determines f .

Definition 2.10: Let (E, F, p, q) be a textile system in which p, q are onto. Define a

system of Wang tiles as follows: T = Tf : f ∈ F 1 where the Tf are formal symbols

indicating a copy of F 1; K is given by K = F 0 t E1; and n, e, s, w : Tf → K are

defined by by n(Tf ) = p(f), s(Tf ) = q(f), e(Tf ) = s(f) and w(Tf ) = r(f) for all

f ∈ F 1.

Pictorially,

fT ( )s f

( )q f

( )r f

( )p f

Figure 2.2 . The system of Wang tiles associated to a textile system.

An important set of examples of systems of Wang tiles which arise from textile sys-

tems:


Example 2.11: Consider the graphs described in Figure 2.3. That is, F 0 = v, w,E0 = u, F 1 = g, h, k, l and E1 = a, b. Define p0(v) = u, q0(v) = u, p0(w) = u

and q0(w) = u, and define p1 : F 1 → E1 and q1 : F 1 → E1 by

p1(g) = p1(k) = a, p1(h) = p1(l) = b;

q1(g) = q1(k) = a, q1(h) = q1(l) = b.

Then p, q : F → E are surjective graph morphisms

v w

l

h

g k:F

uab

:E

Figure 2.3. The textile system of (E, F, p, q).

To see that this is a textile system, we must check that f → (p(f), q(f), r(f), s(f))

is injective on F 1. Since (a, a, v, v), (b, b, w, v), (a, a, w, w) and (b, b, v, w) completely

determine edges g, h, k and l in F 1, (E, F, p, q) is a textile system.

The Wang tiles in the corresponding tiling system are

lT

a

a

v vg

T

b

b

w vh

T

a

a

w wk

T

b

b

wv

Figure 2.4. The set of Wang tile corresponding to the textile system (E, F, p, q).

Definition 2.12: Given a system T, K, n, e, s, w of Wang tiles, a tiling by T is a

function: x : Z2 → T , such that: for every (m,n) ∈ Z2,

(1) e(x(m−1, n)) = w(x(m,n)); and (2) n(x(m,n−1)) = s(x(m,n))(See Figure.2.5).

XT denotes the set of all tilings x : Z2 → T .


( , )x m n( 1, )x m n ( 1, )x m n

( , 1)x m n

( , 1)x m n

Figure 2.5. Tiling x : Z2 → T .

Using Definition 2.12, we can construct a valid tiling for the Wang tiles of Exam-

ple 2.11 as follows. The shaded square Tk indicates where the origin (0, 0) is.

b a

kT

hT

gT

lT

hT

gT

lT

kT

b

b

b

b

b

b

ba

a

a

a

a

a

a

hT

hT

hT

hT

lT

lT

lT

lT

gT

gT

gT

gT

kT

kT

kT

kT

b b b b

b b b b

a

a

a

a

a

a

a

a

Figure 2.6. A tiling x for the textile system of Example 2.11.

Definition 2.13: Given a tiling x : Z2 → T , we define shift map σ on x by

σ(m,n)(x(i, j)) = x(i + m, j + n) for all (m,n), (i, j) ∈ Z2.

This matches up with Definition 2.1. A tiling x is called periodic with period (m,n) ∈Z2\(0, 0) if σ(m,n)(x(i, j)) = x(i, j) for all (i, j) ∈ Z2.

Definition 2.14: Let (T, K, n, e, s, w) be a system of Wang tiles and let x : Z2 → T

be a tiling. Define

Per(x) := (m,n) ∈ Z2 : σ(m,n)(x) = x

We call a pair (m,n) ∈ Per(x) a period of x.


The tiling x of Figure 2.6 is periodic. For example, it has periods (4, 0), (0, 2).

Indeed, Per(x) = (4k1, 2k2) : (k1, k2) ∈ Z2.

Lemma 2.15: Let (T, K, n, e, s, w) be a system of Wang tiles and let x: Z2 → T be

a tiling. Then σ(m,n)(x) is a tiling for each pair (m,n) ∈ Z2, and σ(k,l)(σ(m,n)(x)) =

σ(k+m,l+n)(x) for any pair (k, l), (m,n) ∈ Z2.

Proof. For any pair (i, j) ∈ Z2, we have

(1) e(x(i− 1, j)) = w(x(i, j));

(2) n(x(i, j − 1)) = s(x(i, j));

(3) w(x(i + 1, j)) = e(x(i, j));

(4) s(x(i, j + 1)) = n(x(i, j)).

For (i, j) ∈ Z2, (1) gives

(i) e(σ(m,n)(x)(i − 1, j)) = e(x(i + m − 1, j + n)) = w(x(i + m, j + n)) =

w(σ(m,n)(x)(i, j)). Similarly,

(ii) n(σ(m,n)(x)(i, j − 1)) = s(σ(m,n)(x)(i, j));

(iii) w(σ(m,n)(x)(i + 1, j)) = e(σ(m,n)(x)(i, j));

(iv) s(σ(m,n)(x)(i, j + 1)) = n(σ(m,n)(x)(i, j)).

Thus σ(m,n)(x) is a tiling for each pair (m,n) ∈ Z2. For any (i, j) ∈ Z2,

σ(k,l)(σ(m,n)(x)(i, j)) = σ(k,l)(x(i + m, j + n))

= x(i + m + k, j + n + l) = σ(m+k,n+l)(x)(i, j),

and hence σ(k,l)(σ(m,n)(x)) = σ(k+m,l+n)(x). 2

Let x be the tiling of Example 2.11. The following figures illustrate the tilings

σ(1,0)(x) and σ(0,1)(x). In the diagrams, the shaded square is the one at (0, 0).

0,0x 1,0

x2,0

x 3,0x

4,0x

1,0x2,0

x3,0x

4,1x

3,1x

2,1x1,1

x0,1x1,1

x2,1

x3,1

x

0, 1x 1, 1

x2, 1

x3, 1

x4, 1

x1, 1x

2, 1x3, 1

x

:x

Figure 2.7. The tiling x of (T, K, n, e, s, w).


0,0x 1,0

x2,0

x 3,0x

4,0x

1,0x2,0

x3,0x

4,1x

3,1x

2,1x1,1

x0,1x1,1

x2,1

x3,1

x

0, 1x 1, 1

x2, 1

x3, 1

x4, 1

x1, 1x

2, 1x3, 1

x

(1,0):x

Figure 2.7.(a). The tiling σ(1,0)(x) of (T, K, n, e, s, w).

0,0x

1,0x 2,0

x 3,0x

4,0x

1,0x2,0

x3,0x

4,1x

3,1x

2,1x1,1

x0,1

x1,1x

2,1x

3,1x

0, 1x 1, 1

x2, 1

x3, 1

x4, 1

x1, 1x

2, 1x3, 1

x

(0,1):x

Figure 2.7.(b). The tiling σ(0,1)(x) of (T, K, n, e, s, w).

From Definition 2.13, a tiling x : Z2 → T is periodic with period (m,n) ∈Z2\(0, 0) if and only if: σ(m,n)(x)(i, j) = x(i, j) for all (i, j) ∈ Z2. We can check

that the tiling y illustrated in the system of Example 2.11 is periodic with periods

(0, 2), (2, 0) and (1, 1). The shaded square Th indicates where the origin (0, 0) is (see

Figure 2.8).

hT

lT

hT

hT h

Tl

T lT

lT

b

hTl

Th

T hT

hT

lT

lT

lT

hT

lT

hT

hT

hT

lT

lT

lT

b b b b b b b

b b bb

b b b b

b b b b b b b b

b b b b b b b b

Figure 2.8. The tiling y with period (0, 2), (2, 0) and (1, 1).

Observe that if x : Z2 → T is periodic with period (m,n) ∈ Z2 then x is periodic

with period (km, kn) for all k ∈ Z: if k ≥ 0, then applying formula x(i + m, j + n) =


x(i, j), k times gives

σ(km,kn)(x)(i, j) = x(i + km, j + kn) = x(i, j) for all (i, j) ∈ Z2;

and if k < 0, then applying the above to i′ = i + km, j′ = j + km and k′ = −k gives

the desired formula.

Lemma 2.16: Let (T, K, n, e, s, w) be a system of Wang tiles and let x : Z2 → T be

a tiling. Then Per(x) = (m,n) ∈ Z2 : σ(m,n)(x) = x forms a subgroup of Z2.

Proof. For (m,n), (p, q) ∈ Per(x), we have

σ(m+p,n+q)(x) = σ(m,n)(σ(p,q)(x)) = σ(m,n)(x) = x.

Thus (m + p, n + q) ∈ Per(x). If (m,n) ∈ Per(x), then for (i, j) ∈ Z2, we have

σ−(m,n)(x)(i, j) = x(i−m, j − n) = σ(m,n)(x)(i−m, j − n)

= x(i−m + m, j −m + m) = x(i, j).

Hence (−m,−n) ∈ Per(x). 2

Example 2.17: The tiling system x in Figure 2.8 has periods (2, 0), (0, 2) and (1, 1).

Hence Lemma 2.16 implies that (2b + c, 2a + c) ∈ Per(x) for all a, b, c ∈ Z. We claim

that (a, b) ∈ Z×Z : a+ b ∈ 2Z = Per(x). To see this, suppose first that a+ b ∈ 2Z.

Then a − b ∈ 2Z, and then (a, b) = a(1, 1) + b−a2

(0, 2) ∈ Per(x). Now suppose for

contradiction that (a, b) ∈ Per(x) with a + b /∈ 2Z. Then (a, b), (a, b + 1) ∈ Per(x).

So (a, b + 1)− (a, b) = (0, 1) ∈ Per(x), which is not the case by inspection of x.

We now recall the definition of a 2-graph from [25]. We also describe the skeleton

of a 2-graph and how to recover a 2-graph from its skeleton (see [18]). These 2-graphs

will represent an important class of examples for the rest of the thesis.

Definition 2.18: A small category Λ is a pair of sets Obj(Λ) and Mor(Λ) with maps:

(a) r, s : Mor(Λ) → Obj(Λ);

(b) id : Obj(Λ) → Mor(Λ);


(c) : (µ, ν) ∈ Mor(Λ) × Mor(Λ) : s(µ) = r(ν) → Mor(Λ), such that for all

o ∈ Obj(Λ):

(d) r(ido) = o = s(ido);

(e) λ ids(λ) = λ = idr(λ) λ for all λ ∈ Mor(Λ);

(f) r(µ ν) = r(µ) and s(µ ν) = s(ν) whenever s(µ) = r(ν); and

(g) (µ ν) λ = µ (ν λ).

By convention id(o) is written as ido and (µ, ν) = µ ν is written as µν.

Definition 2.19: Let C and D be categories. A functor F from C to D is a mapping

that:

• associates to each object X ∈ C an object F (X) ∈ D,

• associates to each morphism f : X → Y ∈ C a morphism F (f) : F (X) →F (Y ) ∈ D such that the following two conditions hold:

– F (idX) = idF (X), for every object X ∈ C

– F (g f) = F (g) F (f) for all morphisms f : X → Y , and g : Y → Z.

That is, functors must preserve identity morphisms and composition of mor-

phisms.

Roughly, a small category is a set Λ with partially defined associative multipli-

cation and with local identity elements. Let E be a directed graph, we write En for

the collection of paths of length n in E. Also the set E∗ =∞⋃

n=0

En of all paths is

regarded as a category with objects E0 and composition given by concatenation of

paths. Formally:

Example 2.20: Let E be directed graph, E = (E0, E1, r′, s′). Define category ΛE

as follows:

• Obj(ΛE) = E0,


• Mor(ΛE) = e1e2 . . . en : s′(ei) = r′(ei+1) ∪ E0,

For any λ = e1e2 . . . en ∈ Mor(ΛE)

(a) s(λ) := s′(en) and r(λ) := r′(e1). For v ∈ E0 ⊆ Mor(ΛE), s(v) = r(v) = v.

(b) id : Obj(ΛE) → Mor(ΛE) is given by id(v) = v, and

(c) µ ν = µν for u, v ∈ E∗ with s(µ) = r(ν)

In detail,

1. If µ ∈ E0 and r(λ) = s(µ), then µ = r(λ) and then µλ = λ;

If ν ∈ E0 and s(λ) = r(ν), then ν = s(λ) and λν = λ;

If µ, ν ∈ E0 and s(µ) = r(ν), then µ = ν and µν = µ.

2. If v ∈ E0, then r(v) = v = s(v); and

3. When µ, ν are paths with µ = µ1µ2 . . . µn, ν = ν1ν2 . . . νm, where s(µ) = r(ν),

then µν = µ1µ2 . . . µnν1ν2 . . . νm. Thus s(µν) = s′(νm) = s(ν), and r(µν) =

r′(µ1) = r(µ).

Example 2.21: Let G be a group. We can view G as a category with just one object:

• Mor(G) = G, Obj(G) = e,

• r(g) = s(g) = e for all g ∈ G, and ide = e.

We check that this does indeed define a category:

1. r(ide) = r(e) = e = s(e) = s(ide);

2. g ids(e) = ge = g = eg = idr(e) g;

3. r(g1 g2) = e = r(g1) and s(g1 g2) = e = s(g2) for all g1, g2 ∈ Mor(G);

Example 2.22: G is still a category if G is a monoid, such as M = Nk, which is

defined as Mor(M) = M , Obj(M) = 0 = (0, 0, . . . , 0) ∈ Nk, r(M) = s(M) = 0,(m,n) = m + n and id(0) = 0. We write ei for the generator (0, . . . , 0, 1, 0, . . . , 0)

where 1 occurs in the ith coordinate.


Now, we introduce the definition of a higher-rank graph. We first need some

notations and definitions. For m,n ∈ Nk, we say m ≤ n, if mi ≤ ni for all 1 ≤ i ≤ k.

We write m ∨ n for the coordinate-wise maximum of m and n.

Definition 2.23: ([25]) A graph of rank k or a k-graph (Λ, d) is a countable small

category Λ (with range and source maps r and s respectively) equipped with a functor

d : Λ → Nk, called the degree functor satisfying the following factorization property:

for all λ ∈ Λ and m,n ∈ Nk such that d(λ) = m + n there are unique elements

µ ∈ d−1(m) and ν ∈ d−1(n) such that λ = µν.

The paths in Λ are the morphisms and vertices are the identity morphisms. For

λ ∈ Λ, we call d(λ) = n the degree of λ, and we write Λn := λ ∈ Λ | d(λ) = n. In

this thesis, we are primarily interested in 2-graphs, so k is usually 2. We can visualize

a 2-graph as a directed bicoloured graph with vertex set Λ0 in which the elements of

Λe1 are represented by blue edges from s(λ) ∈ Λ0 to r(λ) ∈ Λ0, and the elements of

Λe2 are represented by red edges. In this thesis, we use black curves to represent blue

edges and dashed curves for red edges. The bicoloured graph is called the skeleton of

Λ. Applying the factorization property to (1, 1) = e1 + e2 = e2 + e1 gives a range and

source preserving bijection between blue-red paths with length 2 and red-blue paths

with length 2. We can visualize a path of degree (1, 1) as a square.

h

g

e

f

Thus the bijection between blue-red paths and red-blue paths discussed above

carries the blue-red path gh to red-blue path ef . The factorization property is deter-

mined by a family of commuting squares C like the one above in which each red-blue

and each blue-red path occurs exactly once.

Notation 2.24: Let Λ be a k-graph, we denote:

1. Λn := d−1(n);


2. r(λ) := idcod(λ) ∈ Λ0 and s(λ) := iddom(λ) ∈ Λ0, for each λ ∈ Λ;

3. For E ⊂ Λ and α ∈ Λ, we write αE := αλ : λ ∈ E, r(λ) = s(α), and

Eα := λα : λ ∈ E, s(λ) = r(α). In particular, for v ∈ Λ0 and n ∈ Nk,

vΛn = λ ∈ Λ : r(λ) = v and d(λ) = n. For v, w ∈ Λ0,

vΛnw = λ ∈ Λ : r(λ) = v, s(λ) = w and d(λ) = n;

4. When m ≤ n ≤ l = d(λ), we write λ(0,m), λ(m,n) and λ(n, l) for the unique

paths of degree m,n−m and l − n such that λ = λ(0,m)λ(m,n)λ(n, l);

5. Let λ(n) = λ(n, n).

Definition 2.25: In a k-graph Λ, a path λ ∈ Λ is a simple cycle if r(λ) = s(λ) and

λ(m) 6= λ(n) for all 0 ≤ m ≤ n ≤ d(λ). We say that a vertex v lies on a simple cycle

λ if λ(n) = v for some n ≤ d(λ).

If d(λ1) = (n1, 0) and d(λ2) = (0, n2) and s(λ1) = r(λ2), then the path λ = λ1λ2 ∈Λ is viewed as a commuting diagram of shape (n1, n2), in which the coordinates of

u, v are (n1, n2) and (0, 0), respectively; and each constituent square belongs to the

commuting square set C. The bottom row of the diagram is the path λ1, and the

right-hand side is the path λ2. The top row is the λ((0, n2), (n1, n2)) and the left-hand

side is λ(0, (0, n2)).

1 1( ) ( ,0)d n

2 2( ) (0, )d n

(0,0)

1 2( , )n n

Figure 2.9: A path of degree n = (n1, n2).

Example 2.26: Let E be a directed graph. Then ΛE = E∗ is a 1-graph where

d : ΛE → N is defined by

d(λ) =

n, if λ = λ1λ2 . . . λn;0, if λ = v.


We check that this is indeed a 1-graph. Firstly, check d is a functor sending

objects to objects. Let µ, ν ∈ Mor(ΛE) such that s(µ) = r(ν). Then

1. If µ ∈ E0, then µ = r(ν), thus µ ν = r(ν) ν = ν. Since d(µ) = 0,

d(µ ν) = d(ν) = d(µ) + d(ν);

2. If ν ∈ E0, then s(µ) = ν, and d(µ ν) = d(µ) = d(µ) + 0 = d(µ) + d(ν) as in

case 1;

3. If µ, ν ∈ E0, then µ = ν = µν. Since d(µ) = d(ν) = d(w) = 0, we have

d(µ ν) = 0 = d(µ) + d(ν);

4. If µ = µ1µ2 . . . µn and ν = ν1ν2 . . . νm, then µν = µ1µ2 . . . µnν1ν2 . . . νm. Thus,

d(µν) = m + n = d(µ) + d(ν).

From above, d is a functor. Suppose d(λ) = m + n where m,n ∈ N, we need to

check that there are unique µ ∈ Em, ν ∈ En such that λ = µ ν.

1. When m = 0, then d(λ) = d(r(λ)λ) = d(r(λ)) + d(λ) = 0 + n = n, so µ = r(λ),

ν = λ works. Now we need to prove the uniqueness. Suppose µ, ν have the

property. In ΛE, λ = µ ν, and d(µ) = 0 and d(ν) = n = d(λ). since

d−1(0) = E0, we have µ = r(ν). Therefore, ν = λ and µ = r(λ) are unique with

this property.

2. When n = 0, then d(λ) = d(λs(λ)) = d(λ)+ d(s(λ)) = m+0 = m, so ν = s(λ),

µ = λ works. The proof of uniqueness of µ, ν is similar to the proof of case 1.

3. When m = n = 0, then d(λ) = d(µ) + d(ν) = 0 + 0 = 0, so λ = µ = ν = v ∈ E0

works. The uniqueness of µ = ν = λ follows from either (1) or (2).

4. When m,n 6= 0, we have λ = λ1λ2 . . . λm+n. So, µ = λ1λ2 . . . λm and ν =

λm+1λm+2 . . . λm+n are clearly the unique paths of Em and En with λ = µν

Now, we introduce the definition of a k-coloured graph and the morphisms be-

tween k-coloured graphs.

Definition 2.27: Let k ≥ 1. A k-coloured graph is a directed graph (E0, E1, r, s)

together with a colour map c : E1 → 1, 2, . . . , k.


Definition 2.28: Given k-graphs (Λ1, d1), (Λ2, d2), k-graph morphism between (Λ1, d1)

and (Λ2, d2) is a degree-preserving functor f : Λ1 → Λ2 such that d2(f(λ)) = d1(λ)

for all λ ∈ Λ1.

Definition 2.29: Given k-coloured graphs (E0, E1, r, s, c) and (F 0, F 1, r′, s′, c′). Then

a coloured-graph morphism ϕ : E → F is a graph morphism such that c′(ϕ(e)) = c(e)

for all e ∈ E1. So a coloured-graph morphism is a graph morphism which preserves

colour.

Definition 2.30: ([61]) Fix k ∈ N and m ∈ Nk. The coloured graph Ek,m has vertices

E0k,m = n ∈ Nk : n ≤ m, and edges E1

k,m = εni : n, n + ei ∈ E0

k,m with structure

maps

r(εni ) = n, s(εn

i ) = n + ei and c(εni ) = i.

For instance, E2,(1,1) has vertices E02,(1,1) = (0, 0), (1, 0), (0, 1), (1, 1); edges E1

2,(1,1) =

ε(0,0)1 , ε

(0,1)1 , ε

(1,0)2 , ε

(0,0)2 .

Thus, E2,(1,1) could be drawn as follows:

(1,0)

2,(1,1)E

(0,1)

(0,0)

(1,1)

(0,0)

1

(0,1)

1

(1,0)

2

(0,0)

2

Figure 2.10. The coloured graph E2,(1,1).

A square in a k-coloured graph E is a coloured graph morphism: ϕ : Ek,ei+ej→ E

for some i 6= j ≤ k.

Definition 2.31: A complete and associative collection of squares for a k-coloured

graph E is a set C of squares in E such that:

1. For each ij-coloured path fg ∈ E2 there is a unique ϕ ∈ C such that ϕ(ε0i ) = f

and ϕ(εeij ) = g; and


2. Write fg ∼ g′f ′ whenever there is a square ϕ such that

ϕ(ε0i ) = f, ϕ(εei

j ) = g, ϕ(ε0j) = g′ and ϕ(ε

ej

i ) = f ′.

If k ≥ 3 and fgh is a tri-coloured path and

fg ∼ g1f1 , f1h ∼ h1f2 g1h1 ∼ h2g2,

gh ∼ h1g1, fh1 ∼ h2f 1 and f 1g1 ∼ g2f 2,

then we require that f2 = f 2 , g2 = g2 and h2 = h2.

Observe when k = 2, the condition (2) is trivial. So for 2-coloured graphs, we

only need to consider condition (1).

Definition 2.32: ([18]) Let Λ be a k-graph. We define a k-coloured graph EΛ and a

collection CΛ of squares associated to Λ as follows. Let EΛ be the k-coloured graph

with E0Λ = v : v ∈ Λ0, E1

Λ =⋃k

i=1f : f ∈ Λei, and c(f) = ci ⇔ d(f) = ei. Define

π : E0Λ → Λ by π(v) = v and π : E1

Λ → Λ by π(f) = f , and extend this to a map

π : E∗Λ → Λ by π(f1 . . . fn) = f1 . . . fn. For distinct i, j ≤ k and λ ∈ Λei+ej define a

coloured-graph morphism ϕλ : Ek,ei+ej→ EΛ by

ϕ0λ(n) = λ(n) and ϕ1

λ(εni ) := λ(n, n + ei).

Let CΛ :=⋃

i<j≤kϕλ : λ ∈ Λei+ej. We call EΛ the skeleton of Λ and CΛ the square

of Λ.

Definition 2.33: Let (E, F, p, q) be a textile system. Form a 2-coloured graph G =

G(E, F, p, q) as follows: G0 = E0, G1 = E1 ∪ F 0 and c : G1 → 1, 2 is defined by

c(e) = 1 for e ∈ E1 ⊆ G1 and c(w) = 2 for w ∈ F 0 ⊆ G1.

We give an example showing how to construct a 2-coloured graph G from a textile

system (E, F, p, q).

Example 2.34: Recall Example 2.11: p1 : F 1 → E1, p1(g) = p1(k) = a, p1(h) =

p1(l) = b; q1 : F 1 → E1, q1(g) = q1(k) = a, q1(h) = q1(l) = b; p0(v) = p0(w) = u,

q0(v) = q0(w) = u; and s(v) = p(v) and r(v) = q(v).


v w

l

h

g k:F

uab

:E

The 2-coloured graph G is:

b:G

v

a

w

u

Figure 2.11. The 2-coloured graph G associated to textile system (E, F, p, q).

The Wang tiles in the corresponding tiling system are

lT

a

a

v vg

T

b

b

w vh

T

a

a

w wk

T

b

b

wv

Figure 2.12. The Wang tile set associated to textile system (E, F, p, q).

The squares ϕ : E2,(1,1) → G correspond to the tiles above, so there are four

squares ϕ1,ϕ2,ϕ3 and ϕ4:

(1). Let ϕ1 : E2,(1,1) → G be defined by ϕ1(ε(0,0)1 ) = a, ϕ1(ε

(0,0)2 ) = v, ϕ1(ε

(1,0)2 ) =

v, ϕ1(ε(0,1)1 ) = a.

(2). Let ϕ2(ε(0,0)1 ) = b, ϕ2(ε

(0,0)2 ) = w, ϕ2(ε

(1,0)2 ) = v, ϕ2(ε

(0,1)2 ) = b.

(3). Let ϕ3(ε(0,0)1 ) = a, ϕ3(ε

(0,0)2 ) = w, ϕ3(ε

(1,0)2 ) = w, ϕ3(ε

(0,1)2 ) = a.

(4). Let ϕ4(ε(0,0)1 ) = b, ϕ4(ε

(0,0)2 ) = v, ϕ4(ε

(1,0)2 ) = w, ϕ4(ε

(0,1)2 ) = b. So ϕ1

corresponds to Tg, ϕ2 to Th, ϕ3 to Tk and ϕ4 to Tl.


a

a

vv

b

b

w

a

a

v

b

b

wwv w

Figure 2.13. Note that the squares set ϕi correspond to Wang tile set associated to

textile system (E, F, p, q).

Now we need to check the 4 squares ϕi : E2,(1,1) → G (i = 1, 2, 3, 4) satisfy the

condition (1) in Definition 2.31.

(1). Let i = 1, j = 2, there are 4 ij-coloured paths in G2 to consider:

(a). Let the ij-coloured path be bw. Then ϕ(ε(0,0)1 ) = b, ϕ(ε

(1,0)2 ) = w, thus there

exists unique ϕ = ϕ4 ∈ C, such that ϕ4(ε(0,0)1 ) = b, ϕ4(ε

(1,0)2 ) = w;

(b). Let the ij-coloured path be aw. Then ϕ(ε(0,0)1 ) = a, ϕ(ε

(1,0)2 ) = w, thus there

exists unique ϕ = ϕ3 ∈ C, such that ϕ3(ε(0,0)1 ) = a, ϕ3(ε

(1,0)2 ) = w;

(c). Let the ij-coloured path be bv. Then ϕ(ε(0,0)1 ) = b, ϕ(ε

(1,0)2 ) = v, thus there

exists unique ϕ = ϕ2 ∈ C, such that ϕ2(ε(0,0)1 ) = b, ϕ2(ε

(1,0)2 ) = v;

(d). Let the ij-coloured path be av. Then ϕ(ε(0,0)1 ) = a, ϕ(ε

(1,0)2 ) = v, thus there

exists unique ϕ = ϕ1 ∈ C, such that ϕ1(ε(0,0)1 ) = a, ϕ1(ε

(1,0)2 ) = v.

(2). Let i = 2, j = 1, there are 4 ij-coloured paths in G2 to consider, and

vb, wa, wb, va are ij-coloured paths in G2:

(a). Let the ij-coloured path be vb. Then ϕ(ε(0,0)2 ) = v, ϕ(ε

(0,1)1 ) = b, thus there

exists unique ϕ = ϕ4 ∈ C, such that ϕ4(ε(0,0)2 ) = v, ϕ4(ε

(1,0)1 ) = b;

(b).Let the ij-coloured path be wa. Then ϕ(ε(0,0)2 ) = w, ϕ(ε

(0,1)1 ) = a, thus there

exists unique ϕ = ϕ3 ∈ C, such that ϕ3(ε(0,0)2 ) = w, ϕ3(ε

(1,0)1 ) = b;

(c). Let the ij-coloured path be wb. Then ϕ(ε(0,0)2 ) = w, ϕ(ε

(0,1)1 ) = b, thus there


(1,0)1 ) = b;

(d). Let the ij-coloured path be va. Then ϕ(ε(0,0)2 ) = v, ϕ(ε

(0,1)1 ) = a, thus there


(1,0)1 ) = b.

Then the squares ϕi : E2,(1,1) → G (i = 1, 2, 3, 4) are complete and associative

collection of squares.

We introduce notation to parameterize the squares in coloured graph G(E, F, p, q)

associated to the textile system (E, F, p, q).


Motivated by the situation of 2-graphs, we associate a set of squares to each

textile system – we do not necessarily expect such a collection to be complete and

associative.

Definition 2.35: Let (E, F, p, q) be a textile system and f ∈ F 1. Define ϕf :

E2,(1,1) → G(E, F, p, q) by

(1). ϕf (ε(0,1)1 ) = p(f); (2). ϕf (0, 1) = r(p(f));

ϕf (ε(0,0)2 ) = r(f); ϕf (0, 0) = r(q(f));

ϕf (ε(0,0)1 ) = q(f); ϕf (1, 1) = s(p(f));

ϕf (ε(1,0)2 ) = s(f). ϕf (1, 0) = s(q(f)).

Then the set of squares can be viewed as follows:

fC ( )s f

( )q f

( )r f

( )p f

C1

f F:

.

3. TEXTILE SYSTEMS AND 2-GRAPHS

To study aperiodicity and entropy of shifts arising from textile systems, we will

describe in this chapter a relationship between some textile systems and the k-graphs

of [26]. The notions of r-path lifting and s-path lifting will be a key ingredient. So

we introduce them next.

Definition 3.1: Let φ : F → E be a surjective graph morphism, we say that:

1) φ has r-path lifting if for every vertex v ∈ F 0 and every edge e ∈ E1 such that

φ(v) = r(e), there exists f ∈ F 1 such that φ(f) = e and r(f) = v. If there is always

a unique f such that φ(f) = e and r(f) = v, we say that φ has unique r-path lifting ;

2) φ has s-path lifting if for every vertex v ∈ F 0 and every edge e ∈ E1 such that

φ(v) = s(e), there exists f ∈ F 1 such that φ(f) = e and s(f) = v. If there is always

a unique f such that φ(f) = e and s(f) = v, we say that φ has unique s-path lifting.

If φ has both r-path lifting and s-path lifting, we say φ has path lifting.

Lemma 3.2: Suppose that p : F → E is a graph morphism. Suppose that p has

r-path lifting. Let n ∈ N, let µ ∈ En and fix v ∈ F 0 such that p(v) = r(µ). Then

there exists λ ∈ F n such that

p(λ) = µ and r(λ) = v. (3.1)

If p has unique r-path lifting, there is a unique λ ∈ F n satisfying (3.1). The symmetric

statement is true of s-path lifting.

Proof. Fix µ ∈ En, and fix v ∈ F 0 such that p(v) = r(µ) ∈ E0. We will inductively

choose edges λi ∈ F 1 such that p(λi) = µi and r(λi) = s(λi−1) for all 1 ≤ i ≤ n. Since

p is a surjective graph morphism, there exists λ1 such that r(λ1) = v and p(λ1) = µ1.

Then p(r(λ1)) = p(v) = r(µ1) = r(µ). Suppose that we have chosen λ1λ2 . . . λn−1

3. Textile systems and 2-graphs 28

such that p(λi) = µi and r(λi+1) = s(λi) for all 1 ≤ i ≤ n − 2. Since p has r-path

lifting, there exists λn ∈ F 1 such that p(λn) = µn and r(λn) = s(λn−1).

If p has unique r-path lifting, there exists unique λ1 such that p(λ1) = µ1 and

r(λ1) = r(λ) = v. Assume that we have chosen λ1λ2 . . . λn−2 such that p(λi) = µi

and r(λi+1) = s(λi) for all 1 ≤ i ≤ n − 2. Since p has unique r-path lifting, there

exists unique λn ∈ F 1 such that p(λn) = µn and r(λn) = s(λn−1).

The symmetric argument using s-path lifting instead r-path lifting shows that if

µ ∈ En and fix v ∈ F 0 such that p(v) = s(µ), there exists ν ∈ F n such that p(ν) = µ

and s(ν) = v. If p has unique r-path lifting, such ν ∈ F n is uniquely chosen. 2

If φ : F → E has unique path lifting, then it is a covering map (see, for example,

Definition 4.7 in [24]). We show next how textile systems with unique path lifting

correspond to 2-graphs.

Lemma 3.3: Let (E, F, p, q) be a textile system, and let G = G(E, F, p, q) be as in

Definition 2.33 and let C = ϕh : h ∈ F 1 be as in Definition 2.35. If p, q have unique

path lifting, then C is a complete and associative collection of squares in G.

Proof. Fix fg ∈ G2 with c(f) = 2, c(g) = 1. We need to prove that there is a

unique h ∈ F 1 with ϕh(ε(0,0)2 ) = f and ϕh(ε

(0,1)1 ) = g. Since c(g) = 1 and c(f) = 2,

we have g ∈ E1 and f ∈ F 0. We say f = v ∈ F 0. Then we have r(g) = s(f) = p(v).

Since p has unique r-path lifting, there exists a unique h ∈ F 1 such that p(h) = g

and r(h) = v, and then ϕh(ε(0,0)2 ) = f and ϕh(ε

(0,1)1 ) = g. If ϕ′h satisfies ϕh′(ε

(0,0)2 ) = f

and ϕh′(ε(0,1)1 ) = g, then r(h′) = r(h) and p(h′) = p(h), so unique r-path lifting forces

h′ = h and hence ϕh′ = ϕh.

The symmetric argument using s-path lifting instead of r-path lifting shows that

if fg ∈ G2 with c(f) = 1 and c(g) = 2, then there is a unique k ∈ F 1 such that

ϕk(ε(0,0)1 ) = f and ϕk(ε

(1,0)2 ) = g. 2

Conversely, we can recover a textile system (E, F, p, q) from a 2-graph Λ. Firstly,

we will give some definitions.

Definition 3.4: The k-graph Λ is row-finite if the set vΛm is finite for each m ∈ Nk

and v ∈ Λ0. Also, Λ has no sources if vΛei 6= ∅ for all v ∈ Λ0 and i ∈ 1, . . . , k;similarly, Λ has no sinks if Λeiu 6= ∅ for all u ∈ Λ0 and i ∈ 1, . . . , k.


Definition 3.5: Let Λ be a 2-graph with no sinks and sources. Form an associated

system (E, F, p, q): Let F 0 = Λe2 and F 1 = Λe1+e2 . Let E0 = Λ0 and E1 = Λe1 ,

and endow E with range and source maps inherited from Λe1 . For λ ∈ F 1, factorize

λ = αµ = νβ where α, β ∈ Λe1 , µ, ν ∈ Λe2 , and define q(λ) = α, p(λ) = β, r(λ) = ν

and s(λ) = µ. For γ ∈ F 0 = Λe2 , q(γ) = r(γ) ∈ Λ0 = E0, p(γ) = s(γ) ∈ Λ0 = E0. So

λ = λ(0, e1)λ(e1, e1 + e2) = q(λ)s(λ)

= λ(0, e2)λ(e2, e1 + e2) = r(λ)p(λ)

in the category Λ.

Lemma 3.6: With Λ and (E, F, p, q) as defined in Definition 3.5, (E, F, p, q) is a

textile system.

Proof. We check that p, q are graph morphisms. Fix λ ∈ F 1. And factorize

λ = αµ = νβ where α, β ∈ Λe1 , µ, ν ∈ Λe2 , and r(λ) = ν and s(λ) = µ. Since

p(r(λ)) = p(ν) = s(ν) = r(β) = r(p(λ)), and since p(s(λ)) = p(µ) = s(µ) = s(β) =

s(p(λ)), then p is a graph morphism. Since q(r(λ)) = q(ν) = r(ν) = r(α) = r(q(λ)),

and since q(s(λ)) = q(µ) = r(µ) = s(α) = s(q(λ)), the map q is also a graph

morphism.

Next we want to show that p, q are surjective. Fix e ∈ E1. Since Λ has no sinks,

there exists f ∈ Λe2 such that s(f) = r(e). Then λ = fe ∈ F 1 satisfies p(λ) = e.

Similarly, since Λ has no sources, there exists g ∈ Λe2 such that r(g) = s(e). Then

µ = eg ∈ F 1 satisfies q(µ) = e. Hence, p, q are surjective.

Fix λ, µ ∈ Λe1+e2 = F 1. Then λ = r(λ)p(λ) = q(λ)s(λ) and µ = r(µ)p(µ) =

q(µ)s(µ). Suppose that (p(λ), q(λ), r(λ), s(λ))=(p(µ), q(µ), r(µ), s(µ)). Then λ =

q(λ)s(λ) = q(µ)s(µ) = µ. So, (E, F, p, q) is a textile system. 2

Definition 3.7: [61] Given a k-coloured graph E and a coloured-graph morphism

ϕ : Ek,m → E, we say that an ij-square ψ in E occurs in ϕ if there exists n ∈ Nk

such that n + ei + ej ≤ m and

ϕ(εni ) = ψ(ε0

i ), ϕ(εn+eij ) = ψ(εei

j ),

ϕ(εnj ) = ψ(ε0

j) and ϕ(εn+ej

i ) = ψ(εej

i ).


If E is a k-coloured graph and C is a complete and associative collection of squares

in E, we say that a coloured-graph morphism ϕ : Ek,m → E is C-compatible if every

square which occurs in ϕ belongs to C.

Theorem 3.8: Let (E, F, p, q) be a textile system such that p, q have unique path

lifting and let G = G(E, F, p, q) be the associated 2-coloured graph. Then there is

a unique 2-graph Λ = Λ(E, F, p, q) such that Λe1 = E1, Λe2 = F 0 and q(f)s(f) =

r(f)p(f) in Λ for all f ∈ F 1. Every element of Λe1+e2 is of the form ϕf for some

f ∈ F 1 (recall definition of ϕf in Definition 2.35).

Proof. By Lemma 3.3, C := ϕh : h ∈ F 1 is a complete and associative collection

of squares in G. By [18, Theorem 4.4],

Λ = ϕ : E2,m → G | m ∈ N2, ϕ is a coloured-graph morphism

is a 2-graph with Λe1 = E1 and Λe2 = F 0. For all f ∈ F 1, we have p(f)s(f) ∼r(f)p(f) in G, as ϕf : E2,(1,1) → G(E, F, p, q) satisfies:

ϕf (ε(0,1)1 ) = p(f), ϕf (ε

(0,0)1 ) = q(f), ϕf (ε

(0,0)2 ) = r(f) and ϕf (ε

(1,0)2 ) = s(f).

So, q(f)s(f) = r(f)p(f) in Λ. 2

Remark 3.9: ([26]) The k-graph which gives us the prototype for a (right) infinite

path is

Ω = Ωk = (m,n) : m,n ∈ Nk : m ≤ n.

with structure maps given by

r(m,n) = m, s(m,n) = n, (l,m)(m,n) = (l, n), d(m,n) = n−m.

where the object space is identified with Nk via (m,m) ↔ m.

Definition 3.10: ([26]) Let Λ be a k-graph and set

ΛΩ = x : Ω → Λ | x is a k-graph morphism.

Note that a k-graph morphism must preserve degree, so that for x ∈ ΛΩ, we have

d(x(m,n)) = n − m. The set ΛΩ is denoted by Λ∞ in [26], but we will consider

right-infinite and bi-infinite paths here, so the notation Λ∞ is potentially confusing.


Definition 3.11: ([26]) Let

∆k = (m,n) : m,n ∈ Zk,m ≤ n;

with structure maps given as in Remark 3.9. It is straightforward to check that (∆, d)

is a k-graph. We use ∆ to form the two-sided infinite path space of a k-graph. If Λ

is a k-graph, let

Λ∆ = x : ∆ → Λ | x is a k-graph morphism.

As in [26], the two-sided infinite path space of Λ has a locally compact Hausdorff

topology: for each n ∈ Zk and λ ∈ Λ set

Z(λ, n) = x ∈ Λ∆ : x(n, n + d(λ)) = λ.

Then the sets Z(λ, n) constitute a basis for the topology on Λ∆. For each n ∈ Zk,

define a map σn : Λ∆ → Λ∆ by

σnΛ(x)(l,m) = x(l + n,m + n).

We omit the subscript Λ on σ if the context in which it is being used is clear.

Lemma 3.12: For each n ∈ N, let n := (n, n, . . . , n) ∈ Nk. The topology on Λ∆ is

generated by the sets Z(µ,−n) where n ranges over N, and µ ranges over Λ2n.

Proof. The sets Z(µ,−n) : n ∈ N, µ ∈ Λ2n are open by definition of the topology

on Λ∆.

Fix a basic open set Z(λ,m) and a point x ∈ Z(λ,m). It suffices to show

that there exists n ∈ N and µ ∈ Λ2n such that x ∈ Z(µ) ⊆ Z(λ,m). Let n =

max|mi|, |mi + di(λ)|, i ≤ k. Let µ = x(−n, n). Then x ∈ Z(µ) by definition of µ.

Moreover, if y ∈ Z(µ), then

y(m,m + d(λ)) = y(−n, n)(m + n,m + n + d(λ)) = µ(m + n,m + n + d(λ))

= x(m,m + d(λ)) = λ (since x ∈ Z(λ,m)). 2

Next we recall the basics of metric spaces. Our objective is to describe a metric

on Λ∆ giving rise to the topology described above.


Definition 3.13: A metric or a distance function on a set Ω is an assignment, to

each pair of points (f, g) ∈ Ω, of a nonnegative real number d(f, g), such that for all

f, g, h ∈ Ω, we have

(a) d(f, g) ≥ 0 and d(f, g) = 0 if and only if f = g;

(b) d(f, g) = d(g, f);

(c) d(f, h) ≤ d(f, g) + d(g, h).

Definition 3.14: For r ∈ N, define [−r, r] ⊆ Z2 by [−r, r] = (p, q) : −r ≤ p, q ≤ r.

Definition 3.15: Let T be a set of Wang tiles and let XT be the shift space of this

set of tiles. Define a function d : XT → [0, 1] as follows: for f, g ∈ XT , define

d(f, g) =

1, if f(0, 0) 6= g(0, 0);inf 1

2r+1 : r ∈ N, f |[−r,r] = g|[−r,r], otherwise.

Lemma 3.16: Given a set T of tiles, d is a metric on XT .

Proof. Fix f, g, h ∈ XT . To prove that d is a metric, we must show that conditions

(a), (b) and (c) of Definition 3.13 hold.

(a). We have d(f, g) ≥ 0 for all f, g ∈ XT by definition. If f = g, then d(f, g) <

2−(r+1) for all r ≥ 0 and so d(f, g) = 0. If f 6= g, there exists r > 0 such that

f |[−r,r] 6= g|[−r,r], then d(f, g) > 2−(r+1) 6= 0, and hence d(f, g) 6= 0.

(b). Both cases in the definition of d are symmetric in f and g.

(c). We need to prove d(f, g) ≤ d(f, h) + d(h, g). This is trivial if f = g, so

suppose that f 6= g.

Case 1: If f(0, 0) 6= g(0, 0), so that d(f, g) = 1.

Subcase 1.1: If f(0, 0) 6= h(0, 0), then d(f, h) = 1 and (a) implies d(g, h) ≥ 0, so

d(f, g) ≤ d(f, h) + d(h, g).

Subcase 1.2: If f(0, 0) = h(0, 0), then f(0, 0) 6= g(0, 0) forces g(0, 0) 6= h(0, 0), so

d(g, h) = 1, and (a) implies that d(f, h) ≥ 0, so d(f, g) ≤ d(f, h) + d(h, g).

Case 2: If f(0, 0) = g(0, 0), define r := maxn ∈ N : f |[−n,n] = g|[−n,n], then

d(f, g) = 12r+1 .


Subcase 2.1: If f(0, 0) 6= h(0, 0), then d(f, h) = 1 and (a) implies d(h, g) ≥ 0, so

d(f, g) ≤ d(f, h) + d(h, g).

Subcase 2.2: Suppose that f(0, 0) = h(0, 0). If f |[−r−1,r+1] 6= h|[−r−1,r+1], then

d(f, h) ≥ 12r+1 = d(f, g). Hence (a) implies that d(f, g) ≤ d(f, h) + d(h, g). If

f |[−r−1,r+1] = h|[−r−1,r+1], then since f |[−r−1,r+1] 6= g|[−r−1,r+1], we have g|[−r−1,r+1] 6=h|[−r−1,r+1]. Thus d(g, h) ≥ 1

2r+1 = d(f, g), and so d(f, g) ≤ d(f, h) + d(h, g). 2

Lemma 3.17: Let (xn)∞n=1 be a sequence of functions xn : Z2 → T . Fix x : Z2 → T .

Then limn→∞

xn = x if and only if for every r ≥ 1, there exists N ∈ N such that n ≥ N

implies that xn|[−r,r] = x|[−r,r].

Proof. (⇐) Fix ε > 0, and let r = max0, dlog21ε−1e. Fix N ∈ N, such that when

n ≥ N , xn|[−r,r] = x|[−r,r]. Then for n ≥ N , d(xn, x) ≤ 12r+1 ≤ ε. Hence lim

n→∞xn = x.

(⇒) Fix r ≥ 1, and let ε = 12r+1 . Since lim

n→∞xn = x, there exists N ∈ N such that

when n ≥ N , d(xn, x) < ε = 12r+1 . Thus when n ≥ N , xn|[−r,r] = x|[−r,r]. 2

Lemma 3.18: Let (xi)∞i=1 be a sequence of tilings. Suppose that lim

i→∞xi = x. Then

x ∈ XT is a tiling.

Proof. In order to prove x is a tiling, we must show that for every (m,n) ∈ Z2,

(i) s(x(m,n + 1)) = n(x(m,n)); and

(ii) w(x(m + 1, n)) = e(x(m,n)).

Fix (m,n) ∈ Z2, and let r = max|m|+1, |n|+1. Since limi→∞

xi = x, Lemma 3.17

implies that there exists N ∈ N such that when i ≥ N , xi|[−r,r] = x|[−r,r]. In particular

(1) xN(m,n + 1) = x(m,n + 1);

(2) xN(m,n) = x(m,n);

(3) xN(m,n− 1) = x(m,n− 1);

(4) xN(m− 1, n) = x(m− 1, n);

(5) xN(m + 1, n) = x(m + 1, n).

That xN is a tiling therefore implies that s(x(m,n + 1)) = s(xN(m,n + 1)) =

n(xN(m,n)) = n(x(m,n)). Similarly, w(x(m+1, n)) = w(xN(m,n)) = e(xN(m,n)) =

e(x(m,n)). 2


Theorem 3.19: Let (T, K, n, e, s, w) be a system of Wang tiles. Then XT is a closed

shift invariant subset of the metric space T Z2

and so is a shift space.

Proof. That XT is shift space invariant follows from Lemma 2.15. That XT is

closed follows from Lemma 3.18. 2

Theorem 3.20: Let (T, k, n, s, e, w) be the Wang tiles associated to the textile sys-

tem (E, F, p, q) and let XT be the shift space of this system of tiles. Suppose p, q

have unique path lifting and Λ is the 2-graph associated to (E, F, p, q). For w ∈ Λ∆,

define h(w) : Z2 → T by h(w)(i, j) = Tf(i,j), where f(i,j) ∈ F 1 is the edge such that

ϕf(i,j)= w((i, j), (i + 1, j + 1)) for (i, j) ∈ Z2. Then the map h : Λ∆ → XT is a

homeomorphism and h σ(m,n)Λ = σ

(m,n)XT

h for all (m,n) ∈ Z2.

Proof. To show h : Λ∆ → XT is a homeomorphism, we must prove (i) that h(w) is

a tiling whenever w ∈ Λ∆; (ii) that h is bijection; and (iii) that h is continuous and

h−1 is continuous.

(i). Fix w ∈ Λ∆. We want to show h(w) is a tiling. For all (i, j) ∈ Z2, we have

e(h(w)(i, j)) = s(f(i,j)) = r(f(i+1,j)) = w(h(w)(i + 1, j)),

which implies that h(w) satisfies the e–w condition.

Since f(i,j) ∈ F 1 for all (i, j) ∈ Z2, we have

n(h(w)(i, j)) = p(f(i,j)) = q(f(i,j+1)) = s(h(w)(i, j + 1))

which implies that h(w) satisfies the n–s condition. Hence, h(w) ∈ XT .

(ii). We first prove that h is surjective. Fix x ∈ XT . For (i, j) ∈ Z2, we have

x(i, j) = Tf(i,j)for some f(i,j) ∈ F 1. Let m ≤ n ∈ Z2. Let G = G(E, F, p, q) be

the coloured graph of Definition 2.33. Define a coloured-graph morphism w(m,n) :

E2,n−m → G by

(1). w(m,n)(ε(i,j+1)1 ) = p(f(i,j)); (2). w(m,n)(i, j + 1) = r(p(f(i,j)));

w(m,n)(ε(i,j)2 ) = r(f(i,j)); w(m,n)(i, j) = r(q(f(i,j)));

w(m,n)(ε(i,j)1 ) = q(f(i,j)); w(m,n)(i + 1, j + 1) = s(p(f(i,j)));

w(m,n)(ε(i+1,j)2 ) = s(f(i,j)). w(m,n)(i + 1, j) = s(q(f(i,j))).


for all 0 ≤ (i, j) ≤ n −m − (1, 1). We claim that w is a degree preserving functor.

w(m,n) is well defined since x is a tiling. From the definition, we have

s(ϕf(i,j)) = s(p(f(i,j))) = s(q(f(i,j+1))) = q(s(f(i,j+1)))

= q(r(f(i+1,j+1))) = r(q(f(i+1,j+1))) = r(ϕf(i+1,j+1)).

Hence, s(ϕf(i,j)) = r(ϕf(i+1,j+1)

). From above, we have w(m,n) ∈ Λn−m, w(n, p) ∈Λp−n and s(w(m,n)) = r(w(n, p)), hence w(m,n)w(n, p) = w(m, p) is a path in Λ of

degree p−m. Moreover, by definition of ∆, (m,n)(n, p) = (m, p) and d(m, p) = p−m.

Hence, w is a degree preserving functor. For any (i, j) ∈ Z2, there exists f(i,j) ∈ F 1

such that h(w)((i, j), (i + 1, j + 1)) = Tf(i,j)= x(i, j), so h(w) is surjective.

We prove that h(w) is injective. Suppose that (i, j) ∈ Z2, satisfy h(w)((i, j), (i +

1, j + 1)) = Tf(i,j)= Tf ′

(i,j). Since h(w)((i, j), (i + 1, j)) = q(f(i,j)) = q(f ′(i,j)) and

h(w)((i, j), (i, j + 1)) = r(f(i,j)) = r(f ′(i,j)), and since q has unique r-path lifting, we

have f(i,j) = f ′(i,j). So h is injective. For all (m,n), (i, j) ∈ Z2 and w ∈ Λ∆,

(h σ(m,n)Λ )(w)(i, j) = h(σ

(m,n)Λ (w)(i, j)) = h(w(m + i, n + j))

= Tf(m+i,n+j)= σ

(m,n)XT

(Tf(i,j)) = σ

(m,n)XT

(h(w)(i, j))

= (σ(m,n)XT

h)(w)(i, j).

Hence, h σ(m,n)Λ = σ

(m,n)XT

h.

(iii). We recall the topologies on XT and Λ∆. For a ∈ XT , r ∈ N, define

Br(a) = b ∈ XT : d(a, b) < 2−(r+1) = b ∈ XT : a|[−r,r] = b|[−r,r]. (3.2)

For y ∈ Λ∆, set

Br(y) = Z(y(−r, r),−r). (3.3)

By Definition 3.11, Br(y) is open.

Let G ⊆ XT be open. To prove that h is continuous, it is suffices to show that

h−1(G) is an open set in Λ∆. Fix w ∈ h−1(G). Then h(w) ∈ G. Since G is open,

there exists r > 0 such that Br(h(w)) ⊆ G, where Br(h(w)) is defined as (3.2).

We claim that h−1(Br(h(w))) = Br(w). To prove this claim, we must show (a)

h−1(Br(h(w))) ⊆ Br(w) and (b) Br(w) ⊆ h−1(Br(h(w))).


In (a), fix x ∈ h−1(Br(h(w))), we want to show x ∈ Br(w). Now, h(x) ∈Br(h(w)), so h(x)|[−r,r] = h(w)|[−r,r]. Hence h(x)(i, j) = h(w)(i, j) by definition

of h. So x((i, j), (i + 1, j + 1)) = w((i, j), (i + 1, j + 1)). Then x(−r, r) = w(−r, r),

giving x ∈ Br(w).

In (b), fix y ∈ Br(w). We want to show that y ∈ h−1(Br(h(w))) or h(y) ∈Br(h(w)). Since y ∈ Br(w), then y(−r, r) = w(−r, r), hence y((i, j), (i + 1, j + 1)) =

w((i, j), (i + 1, j + 1)). Then by definition of h, we have h(y)|[−r,r] = h(w)|[−r,r]. So

h(y) ∈ Br(h(w)).

Hence w ∈ Br(w) = h−1(Br(h(w))) ∈ h−1(G), and so h−1(G) is open and h is

continuous.

The set Λ∆ is a closed subset of Πn∈Z2

(Λe1 tΛe2), Tychonoff’s theorem implies that

the product space Πn∈Z2

(Λe1 t Λe2) is compact, so Λ∆ is compact. Similarly, XT is

compact. A continuous bijection between two compact sets has continuous inverse

function, so h−1 is continuous. Hence h is a homeomorphism. 2

Next, we recall the notation of cofinality of 2-graphs.

Definition 3.21: ([29]) Let Λ be a k-graph. We say that Λ is cofinal if for any

v, w ∈ Λ0, there exists N ∈ Nk such that for all α ∈ wΛN , there exists β ∈ vΛs(α).

The cofinality of directed graphs is different from k-graphs. Following [54], a

directed graph E with no sources is cofinal if for every x ∈ F∞ and v ∈ E0, there is

i ∈ N such that there is a path from r(xi) to v. In [29, Theorem 5.1], Lewin and Sims

proved that a graph E is cofinal if and only if the 1-graph E∗ is cofinal.

Lemma 3.22: Let (E, F, p, q) be a textile system. Suppose that p, q have unique

path lifting and let Λ = Λ(E, F, p, q) be the associated 2-graph. If E is cofinal, then

Λ is cofinal.

Proof. Fix v, w ∈ E0 = Λ0. By definition of cofinality of E, there exists N1 ∈ Nsuch that for any α ∈ wEN1 , there exists β ∈ vE∗s(α). By definition of Λ(E, F, p, q),

each path α ∈ wEN1 is also a path in wΛ(N1,0). Now β belongs to vΛ∗s(α) as well.

Hence, Λ is cofinal. 2


Definition 3.23: A directed graph H is called connected if the underlying undirected

graph H is connected. It is called strongly connected if H contains a directed path

from u to v and a directed path from v to u for every pair of vertices u, v. The

strongly connected components of H are the maximal strongly connected subgraphs

of H.

The converse of Lemma 3.22 fails. To see this, we present an example of textile

system with unique path lifting such that the associated 2-graph Λ is cofinal, but the

graph E is not.

Example 3.24: Let E, F be the directed graphs pictured below.

1u

1f:F

:E

2u

2f

3u

3f

1v

1e

2v

2e

Define p, q : F → E by

p(u1) = v1, p(u2) = v1, and p(u3) = v2;

q(u1) = v2, q(u2) = v1, and q(u3) = v1.

Then we obtain a tiling system with skeleton (E, F, p, q). Since p, q have unique

r-path lifting and s-path lifting, we can form 2-graph G = G(E, F, p, q),

1v

2v:G

1u

3u

1e

2e

2u

Figure 3.1. The associated 2 coloured-graph G = G(E, F, p, q).


Since E is disconnected, E is not cofinal. Now G is strongly connected as a directed

graph: vGw 6= ∅ for all v, w ∈ G0. In [33, Proposition 4.5], Maloney and David

proved that if G is strongly connected, then G is cofinal.

Lemma 3.25: Let (E, F, p, q) be a textile system such that p, q have unique path

lifting. Let Λ = Λ(E, F, p, q) be the associated 2-graph. The red graph (the subgraph

consisting of all red edges Λe2 in Λ) of Λ is strongly connected if and only if there

exists n ∈ N such that for any u ∈ F 0,⋃

m≤n

(p−1 q)m(u) = F 0.

Proof. (⇒) Suppose the dashed graph of Λ is strongly connected. Fix u ∈ F 0, then

u ∈ Λe2 and p(u) ∈ Λ0. Fix v ∈ F 0. Then v ∈ Λe2 and q(v) ∈ Λ0. Since the dashed

graph of Λ is strongly connected, there exist paths α, β ∈ Λ such that: α ∈ Λ(0,n1)

with r(α) = q(v), s(α) = p(u).

Suppose α = v · u1 · u2 · . . . · un1−2 · u. By definition of G,

p(v) = q(u1)

⇒ v ∈ (p−1 q)(u1)

⊆ (p−1 q)(p−1 p)(u1)

= (p−1 q)(p−1)(p(u1))

= (p−1 q)(p−1)(q(u2))

= (p−1 q)(p−1 q)((u2)

= (p−1 q)2(u2)

. . .

⊆ (p−1 q)(n1−2)(un1−2)

⊆ (p−1 q)(n1−1)(u)

Thus, v ∈ (p−1 q)n(u), n = n1 − 1. Since v ∈ F 0 was arbitrary, for any v, there

exists n(u, v) ∈ N , such that v ∈ (p−1 q)n(u,v)(u). Now n(u) := maxv∈F 0

n(u, v) satisfies⋃

m≤n(u)

(p−1 q)m(u) = F 0. Then n := maxu∈F 0

n(u) does the job.

(⇐) Suppose there exists n ∈ N such that for any u ∈ F 0,⋃

m≤n

(p−1 q)m(u) = F 0.

Fix (u, v) ∈ F 0 × F 0. There exists n1, n2 ≤ n such that v ∈ (p−1 q)n1(u) and

u ∈ (p−1 q)n2(v). Since v ∈ (p−1 q)n1(u), there is a sequence of vertices vi ∈(p−1 q)i(u) ⊆ F 0 such that v0 = u, vn1 = v and vi ∈ (p−1 q)(vi−1) for all 0 ≤ i ≤ n1.

By construction of G, there is a red path α = vn1 · . . . · v1 · v0 = v · . . . · v1 · u ∈ Λ(0,n1)


with r(α) = q(v), s(α) = p(u). Since p and q are surjective, it follows that the dashed

graph of Λ is strongly connected. 2

Example 3.26: We next determine the number of the value of n in Example 3.24:

(1). For u1. (p−1 q)(u1) = u3, and (p−1 q)2(u1) = u1, u2. Thus, when

n = 2,⋃

m=1,2

(p−1 q)m(u1) = u1, u2, u3 = F 0;

(2). For u2. (p−1 q)(u2) = u1, u2, and (p−1 q)2(u2) = u1, u2, u3.(3). For u3. (p−1 q)(u3) = u1, u2, and (p−1 q)2(u3) = u1, u2, u3.

Hence, the value of n in Example 3.24 is 2.

In the following, we write (Λe1)∗ for the 1-graph with vertices Λ0and edges Λe1 .

Definition 3.27: Let (E, F, p, q) be a textile system. Suppose that p, q have unique

path lifting, and let Λ = Λ(E, F, p, q) be the associated 2-graph. Let C,D ⊆ Λ0 be

strongly connected components of (Λe1)∗. We say that C ¹ D if there is a path in

(Λe1)∗ from D to C. Say C is a sink component in (Λe1)∗, if there exists no D such

that D ¹ C; and say C is a source component in (Λe1)∗, if there exists no D such that

C ¹ D.

Remark 3.28: “¹” is a partial order on the set of strongly connected components.

Lemma 3.29: Let (E, F, p, q) be a textile system with E, F possibly infinite, and

such that p, q have unique path lifting and let Λ = Λ(E, F, p, q) be the associated

2-graph. Then Λ is strongly connected if and only if whenever C and D are strongly

connected components in E, there exist C0 ¹ C and D ¹ D0, and there exist n ∈ Nand u ∈ F 0, such that p(u) ∈ C0 and ((q p−1)n p)(u) ∩ D0

0 6= ∅.

Proof. (⇒) Suppose that Λ is strongly connected. Fix a pair of strongly connected

components C and D in E. Since Λ is strongly connected, there exists a path λ with

s(λ) ∈ C and r(λ) ∈ D. Factorize λ = λbλr where d(λb) = (d(λ)1, 0) and d(λr) =

(0, d(λ)2). Then, s(λb) = s(λ) ∈ C. Since Λe1 = E1, there exists C0 such that C0 ¹ Cand r(λb) ∈ C0. Similarly, there exists D0 such that D ¹ D0, s(λr) = r(λb) ∈ C0

and r(λr) ∈ D0. Thus, λr is a red path from C0 to D0. Then, there exist n ∈ N and

u ∈ F 0, such that p(u) ∈ C0 and ((q p−1)n p)(u) ∩ D00 6= ∅.


(⇐) Suppose for each pair of strongly connected components C and D in E, there

exist C0 ¹ C and D ¹ D0, and there exist n ∈ N and u ∈ F 0, such that p(u) ∈ C0

and ((q p−1)n p)(u) ∩ D00 6= ∅. Let C and D be strongly connected components of

E such that C ¹ D. We claim that:

(i). For each v ∈ D, there exists µ0 ∈ E∗ such that s(µ0) = v and r(µ0) ∈ C;

likewise

(ii). For each w ∈ C, there exists ν0 ∈ E∗ such that r(ν0) = w and s(ν0) ∈ D.

To prove (i), fix v ∈ D0. Since C ¹ D, there exists a path α0 ∈ E∗ such that

r(α0) ∈ C and s(α0) ∈ D. Since D is a strongly connected component, there exists

a path β0 ∈ E∗ such that r(β0) = s(α0) and s(β0) = v. Hence µ0 := α0β0 satisfies

s(µ0) = s(α0β0) = v and r(µ0) = r(α0β0) ∈ C. The proof of (ii) is similar.

Now fix v ∈ C0 and w ∈ D0. Fix n ∈ N and u ∈ F 0, such that p(u) ∈ C0 and

((q p−1)n p)(u) ∩ D00 6= ∅. By hypothesis, there exists a red path γ ∈ (Λe2)∗ from

C0 to D0 such that r(γ) ∈ D00 and s(γ) ∈ C0

0 . Since C0 ¹ C, there exists α ∈ E∗ such

that r(α) ∈ C00 and s(α) ∈ C. Since C is strongly connected, there exists µ ∈ E∗ such

that r(µ) = s(α) ∈ C0 and s(µ) = v. Similarly, for D and D0, there exist ν ∈ E∗ and

β ∈ E∗ such that r(ν) = w, s(ν) = r(β) ∈ D0 and s(β) ∈ D00. Since C0 and D0 are

strongly connected components, there exist blue paths η and ξ such that s(η) = r(α),

r(η) = s(γ), r(γ) = s(ξ) and r(ξ) = s(β). Hence νβξγηαµ is a path from v to w. So

Λ is strongly connected. 2

Lemma 3.29 provides us a necessary and sufficient condition for cofinality of Λ.

When E0 is finite, we obtain more special condition for it.

Corollary 3.30: Let (E, F, p, q) be a textile system such that p, q have unique path

lifting and let Λ = Λ(E, F, p, q) be the associated 2-graph. Suppose that E is finite.

Then Λ is cofinal if and only if for each sink component C and each source component

D in E, there exist u ∈ F 0 and n ∈ N such that p(u) ∈ C and ((qp−1)np)(u)∩D 6= ∅.

Proof. (⇒) This implication follows from Lemma 3.29.

(⇐) Suppose for each sink component C and each source component D there exist

u ∈ F 0 and n ∈ N such that p(u) ∈ C and ((q p−1)n p)(u) ∩ D 6= ∅.


Fix v, w ∈ Λ0. Let Cv be the strongly connected component of E containing v,

and let Cw be the strongly connected component of E containing w. We claim that:

(a). there exists a blue path µ with s(µ) = v and r(µ) ∈ C for some sink

components C; and

(b). there exists a blue path ν with r(ν) = w and s(ν) ∈ D for some source

components D.

To prove (a), fix v ∈ Cv. Suppose that (a) does not hold, and seek a contradiction.

Then, Cv is not a sink component, so there exists a strongly connected component C1

such that C1 ≺ Cv. Moreover, C1 is not a sink component because (a) does not hold.

Hence there exists C2 such that C2 ≺ C1 ≺ Cv. Moreover, C2 is not a sink component.

Following this way, we obtain . . . ≺ Ci ≺ Ci−1 . . . ≺ C1 ≺ Cv such that no Ci is a sink

component. By Remark 3.28, Ci 6= Cj if i 6= j ∈ N, so∞⋃i=1

C0i is infinite, contradictory

that E0 is finite. Thus, (a) holds. A similar argument gives (b).

By hypothesis, there exists α ∈ (Λe2)∗ such that s(α) ∈ C and r(α) ∈ D. Since

C,D are strongly connected components in E, we can find η ξ ∈ E∗ such that:

s(η) = r(µ), r(η) = s(α), s(ξ) = r(α) and r(ξ) = s(ν). Thus, νξαηµ is a path from

v to w. Thus, Λ is strongly connected and hence cofinal. 2

Now, we will study the aperiodicity of 2-graphs.

Definition 3.31: [55] Let Λ be a row-finite k-graph with no sources. We say that Λ

has no local periodicity at v ∈ Λ0 if for each pair m 6= n ∈ Nk, there is a path λ ∈ vΛ

such that d(λ) ≥ m ∨ n and

λ(m,m + d(λ)− (m ∨ n)) 6= λ(n, n + d(λ)− (m ∨ n)).

The k-graph Λ is said to be aperiodic if it has no local periodicity at v for every

v ∈ Λ0.

Definition 3.32: Let E be a directed graph and let α ∈ E∗ be a cycle. An edge e is

an entry to α if there exists i such that r(e) = r(αi) and e 6= αi.


lifting. Fix w ∈ F 0. Suppose that |r−1E (q(w))| = 1. Then |r−1

F (w)| = |r−1E (p(w))| = 1.


Proof. Let e be the unique edge of r−1E (q(w)). Fix f, g ∈ r−1

F (w). Then r(q(f)) =

q(r(f)) = q(w). So, q(f) ∈ r−1E (q(w)), forcing q(f) = e. Similarly, q(g) = e. Since

q has unique path lifting, f = g. Now, suppose that h, k ∈ r−1E (p(w)). By unique

path lifting, there exist f, g ∈ r−1F (w) such that p(f) = h and p(g) = k. From above,

f = g. Hence, h = k. Thus, |r−1F (w)| = |r−1

E (p(w))| = 1. 2

Notation 3.34: In order to avoid the confusion with our notation Λ∞ for infinite

paths in k-graphs, we will give notation of infinite paths in directed graphs as follows:

Let E be a directed graph. Write XE the set of bi-infinite paths in E such that

XE := . . . e−2e−1e0e1e2 . . . : ei ∈ E1, s(ei) = r(ei+1). Moreover, we will write

X+E := e1e2 . . . : ei ∈ E1, s(ei) = r(ei+1) and X−

E := . . . e−2e−1 : ei ∈ E1, s(ei) =

r(ei+1) the sets of one-sided infinite paths in E. What we are denoting, X+E is often

written E∞ in the literature.

If x = e1e2 . . . ∈ X+E is an infinite path in a 1-graph, then we define r(x) = r(e1)

and for n ∈ N, σn(x) = en+1en+2 . . . ∈ X+E .


lifting. Let Λ = Λ(E, F, p, q) be the associated 2-graph. If the 2-graph Λ is aperiodic

then for each v ∈ E0, there exists x ∈ X+E such that r(x) = v and σm(x) 6= σn(x)

whenever m 6= n ∈ N.

Proof. We argue by contrapositive. Suppose that there is a vertex v ∈ E0 such

that every x ∈ X+E such that r(x) = v is periodic.

By [26], there exists v ∈ E0 such that vX+E = µ∞ for some cycle µ ∈ E∗.

Denote µ := e1e2 . . . en. Then |r−1(r(ei))| = 1 in E for all i. Fix w ∈ F 0 such that

q(w) = r(µ). Lemma 3.33 implies that |r−1F (w)| = 1, say r−1

F (w) = f1. Another

application of Lemma 3.33 implies that |r−1F (s(f1))| = 1, say r−1

F (s(f1)) = f2.Continuing this way, we see that x1 := f1f2 . . . is the unique infinite path coming into

p(w). For each ei, |r−1(r(ei))| = 1, and so |r−1(r(x1i ))| = 1 for all i by Lemma 3.33.

Since E0 is finite, there is a unique cycle α1 ∈ E∗ such that x1 = α1α1α1 . . . and

α1 has no entrance. Hence (p q−1)(µm1) = αn11 for some m1, n1 ∈ N. Iterating this

construction, we obtain a sequence of simple cycles αi with no entrance such that


xi+1 := αi+1αi+1αi+1 . . . satisfies xi+1 = p q−1(xi) = (p q−1)i+1(µ∞). Since E0 is

finite, there exist distinct i, j such that r(αi) = r(αj). Since the αi have no entrance

and are simple, this forces αi = αj. By unique path lifting, this forces αi+1 = αj+1 and

by induction αi+n = αj+n for all n. Let N := (lcm(|αi|, |αi+1|, . . . , |αj−1|), 0) ∈ N2.

Fix λ ∈ r(µ)Λ with d(λ) > N . Let m := 0 and n := N , so m ∨ n = N . Then

λ(m,m + d(λ)− (m ∨ n)) = λ(0, d(λ)−N) = λ(N, d(λ))

= λ(n, n + d(λ)− (m ∨ n)).

Hence Λ has locally periodicity (0, N) at w and so is not aperiodic. 2

There is an obvious dual textile system obtained by interchanging the roles of

north, south maps vertically and east, west maps horizontally. Given a textile system

(E, F, p, q), its dual textile system is obtained by interchanging the pairs of maps

(p, q) and (s, r). This is an important ingredient in Nasu’s work [43], [46].

Definition 3.36: Let (E, F, p, q) be a textile system. The dual system (G,H, π, φ)

[13] is defined as follows: G0 = E0, G1 = F 0, rG = q and sG = p; and H0 = E1,

H1 = F 1, rH = q and sH = p. Define maps π, φ : (H0, H1, rH , sH) → (G0, G1, rG, sG)

by:

π(e) = rE(e) if e ∈ H0 = E1 and π(f) = rF (f) if f ∈ H1 = F 1;

φ(e) = sE(e) if e ∈ H0 = E1 and φ(f) = sF (f) if f ∈ H1 = F 1.

Lemma 3.37: Let (E, F, p, q) be a textile system. Then the dual system (G,H, π, φ)

is also a textile system. For e ∈ E1 = H0 and v ∈ F 0 = G1, we have:

(a) q−1(e) ∩ r−1F (v) = π−1(e) ∩ r−1

H (e); (b) q−1(e) ∩ s−1F (v) = φ−1(e) ∩ r−1

H (e);

(c) p−1(e) ∩ r−1F (v) = π−1(e) ∩ s−1

H (e); (d) p−1(e) ∩ s−1F (v) = φ−1(e) ∩ s−1

H (e).

Proof. We want first to show that that (G,H, π, φ) is a textile system and has path

lifting. We need to check that π, φ are graph morphisms. Fix h ∈ H1 = F 1. Fix

v = rF (h), and fix e such that e = rH(h) = q(h) ∈ H0 = E1 and π(e) = rE(e) = q(v)

where v ∈ F 0. Then we have

π(rH(h)) = π(e) = π(q(h)) = rE(e) = q(v)

= q(rF (h)) = q(π(h)) = rG(π(h)).


Hence π(rH(h)) = rG(π(h)). With same argument, we can show that π(sH(h)) =

sG(π(h)). So π is a graph morphism. Similarly, φ(rH(h)) = rG(φ(h)) and φ(sH(h)) =

sG(φ(h)) for all h ∈ F 1, so φ is also a graph morphism.

We now show that π, φ are surjective and have path lifting. Fix v ∈ G1 = F 0 such

that rG(v) = q(v) and sG(v) = p(v). Fix e ∈ H0 = E1 such that π(e) = rE(e) = q(v).

Since q has r-path lifting in (E, F, p, q), there exists h ∈ F 1 such that q(h) = e and

rF (h) = v. Then h ∈ H1 and rH(h) = e and π(h) = v and q(h) = e. So π is

surjective and has r-path lifting. A similar argument shows that π has s-path lifting,

and symmetric arguments show that φ is surjective and has path lifting.

Since (E, F, p, q) is a textile system, f → (p(f), q(f), r(f), s(f)) is injective in F 1.

Fix h ∈ H1 such that h = f ∈ F 1. We have:

(π(h), φ(h), rH(h), sH(h)) = (π(f), φ(f), rH(h), sH(h))

= (rF (f), sF (f), q(h), p(h))

= (rF (f), sF (f), q(f), p(f)).

Since f → (p(f), q(f), r(f), s(f)) is injective in F 1, h → (π(h), φ(h), rH(h), sH(h)) is

injective on H1. Hence (G,H, π, φ) is a textile system and has path lifting.

To prove (a), fix e ∈ E1 = h0 and v ∈ F 0 = G1 such that π(e) = rG(v).

Then q−1(e) ∩ r−1F (v) is the set of f ∈ F 1 such that rF (f) = v and q(f) = e.

Simultaneously, π−1(e) ∩ r−1H (e) is the set of h ∈ H1 = F 1 such that rH(h) =

q(h) = e and π(h) = rF (h) = v. Since H1 = F 1, q−1(e) ∩ r−1F (v) = π−1(e) ∩ r−1

H (e),

and (a) is confirmed.

For (b), fix e ∈ E1 = h0 and v ∈ F 0 = G1 such that π(e) = sH(v). We have

that q−1(e) ∩ s−1F (v) is the set of f ∈ F 1 such that sF (f) = v and q(f) = e.

Simultaneously, φ−1(e) ∩ r−1H (e) is the set of h ∈ H1 = F 1 such that rH(h) =

q(h) = e and φ(h) = sF (h) = v. Since H1 = F 1, q−1(e) ∩ s−1F (v) = φ−1(e) ∩ r−1

H (e),

and (b) is confirmed.

Similar arguments establish (c) and (d). 2

Remark 3.38: Equation (a) in Lemma 3.37 implies that q has [unique] r-path lifting

if and only if π has [unique] r-path lifting. So (a) and (b) say q has [unique] path


lifting if π, φ both have [unique] lifting. In particular, the system (E, F, p, q) has

[unique] path lifting if and only if the system (G,H, π, φ) has [unique] path lifting.


lifting. Let Λ = Λ(E, F, p, q) be the associated 2-graph and Λ(G,H, π, φ) be the 2-

graph associated to the dual system (G,H, π, φ). Define F : N2 → N2 by F(m,n) =

(n,m). Then (Λ(G,H, π, φ),F dΛ(G,H,π,φ)) ∼= (Λ(E, F, p, q), dΛ(G,H,π,φ)).

Proof. By Lemma 3.37 and Remark 3.38, we have (G,H, π, φ) is a textile system

and has unique path lifting. Theorem 3.8 yields one associated 2-graph Λ(G,H, π, φ)

satisfying

Λ0(G,H, π, φ) = Λ0(E, F, p, q), (3.4)

Λ(1,0)(G,H, π, φ) = Λ(0,1)(E, F, p, q), and (3.5)

Λ(0,1)(G,H, π, φ) = Λ(1,0)(E, F, p, q). (3.6)

If e, e′ ∈ Λ(0,1)(G,H, π, φ) = Λ(1,0)(E, F, p, q) and f, f ′ ∈ Λ(1,0)(G,H, π, φ) =

Λ(0,1)(E, F, p, q), then ef = f ′e′ in Λ(G,H, π, φ) if and only if ef = f ′e′ in Λ(E, F, p, q).

Hence Λ(m,n)(G,H, π, φ) = Λ(n,m)(E, F, p, q) and this identification preserves compo-

sition. Since F is an automorphism of N2, then F dΛ(G,H,π,φ) is a functor from

Λ(G,H, π, φ) to N2, and has the factorization property. Hence

(Λ(G,H, π, φ),F dΛ(G,H,π,φ)) ∼= (Λ(E, F, p, q), dΛ(G,H,π,φ)). 2

Proposition 3.40: Let (E, F, p, q) be a textile system such that p, q have unique

path lifting. Let Λ = Λ(E, F, p, q) be the associated 2-graph. If the 2-graph Λ is

aperiodic, then for each vertex v ∈ F 0, there exists an aperiodic infinite sequence

x1, x2, . . . , xi, . . . such that xi ∈ F 0 and xi ∈ (q−1 p)i(v).

Proof. By Lemma 3.39, we have that

(Λ(G,H, π, φ),F dΛ(G,H,π,φ)) ∼= (Λ(E, F, p, q), dΛ(G,H,π,φ)).

Since Λ(E, F, p, q) is aperiodic, Λ(G,H, π, φ) is also aperiodic. By Lemma 3.35, if

Λ(G,H, π, φ) is aperiodic, then for each v ∈ G, there exists x ∈ G∞ such that

r(x) = v and σm(x) 6= σn(x) whenever m 6= n ∈ N. By definition of (G,H, π, φ),


the infinite path x with r(x) = v ∈ F 0 satisfies that r(x) = r(x1) = v = q(x1),

r(xi) = q(xi), s(xi) = p(xi) and p(xi) = q(xi+1) for all xi ∈ G1 = F 0. Therefore

xi = (q−1 p)i(v) for all i ≥ 1. Since σm(x) 6= σn(x) for any m 6= n ∈ N, the sequence

(v, x1, x2, . . . , xi, . . .) is aperiodic with xi ∈ (q−1 p)i(v). 2

However, given a textile system (E, F, p, q), if there exists an aperiodic tiling, we

can not conclude that for every vertex in E, there exists an aperiodic infinite path in

E coming into it.

Example 3.41: Let E, F be the graphs below.

1u

1f

2f

2u3

f 4f

5f:F

2v

:E1

v

Define graph morphisms p, q : F → E by:

p(u1) = v2, q(u1) = v1, p(u2) = q(u2) = v2;

p(f1) = β, p(f2) = γ, p(f3) = β, p(f4) = γ, p(f5) = γ;

q(f1) = α, q(f2) = α, q(f3) = β, q(f4) = γ, q(f5) = γ.

Then one can check that the following is a valid aperiodic tiling of Example 3.41

(the shaded square Tf1 indicates where the origin (0, 0) is).


1fT

1fT

1fT

2fT

2fT

2fT

2fT

2fT

2fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

3fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

4fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

5fT

Figure 3.2: A valid aperiodic tiling of Example 3.41.

However at v1 ∈ E0, there only exists a loop α, so there only exists one path (α)∞

which is a periodic path coming into v1.

We conclude this chapter by investigating how the dual graph construction of [2]

interacts with textile systems.

Definition 3.42: [2] Let (Λ, d) be a k-graph and let p ∈ Nk. Let pΛ = λ ∈ Λ :

d(λ) ≥ p. Define range and source maps on pΛ by rp(λ) = λ(0, p), and sp(λ) =

λ(d(λ) − p, d(λ)) for all λ ∈ pΛ, and define composition by λ p µ = λµ(p, d(µ)) =

λ(0, d(λ) − p)µ whenever sp(λ) = rp(µ). Finally, define a degree map dp on pΛ by

dp(λ) = d(λ)− p for all λ ∈ pΛ.

It is proved in [2] that pΛ is a k-graph.

Definition 3.43: let (E, F, p, q) be a textile system such that p, q have unique path

lifting. For any e1e2 ∈ E2, define r(e1e2) = e1 and s(e1e2) = e2; and for any f1f2 ∈f 2, define r(f1f2) = e1 and s(f1f2) = f2. Define (E, F , p, q) as E = (E1, E2, r, s),


F = (F 1, F 2, r, s). Define p, q as: for f1f2 ∈ F 1, p(f1f2) = p(f1)p(f2) ∈ E1 and

q(f1f2) = q(f1)q(f2) ∈ E1.

That p, q have path lifting implies that p, q are graph morphisms with path lifting

(see Theorem 3.48).

Example 3.44: Let (E, F, p, q) be the textile system of Example 2.11. Then the

associated system (E, F , p, q) is given by: E0 = a, b, E1 = aa, ab, ba, bb; F 0 =

g, h, l, k, F 1 = gg, gl, hg, kh, kk, lk, lh, hl;p(gg) = p(kk) = aa, p(gl) = p(kh) = ab, p(lk) = p(hg) = ba, and p(lh) = p(hl) = bb;

and

q(gg) = q(kk) = aa, q(gl) = q(kh) = ab, q(lk) = q(hg) = ba, and q(lh) = q(hl) = bb.

a b

ab

ba

aabb

ˆ :E

gk

l

h

gg kkˆ :F

hg

gl

kh

lk

hl lh

p q

Figure 3.3. The configuration of (E, F , p, q).

The 2-graphs Λ(E, F , p, q) and (1, 0)Λ(E, F, p, q) are as follows:

ab

ab

ba

aa bbgk h l

Figure 3.3(a). The configuration of Λ(E, F , p, q).


a b

ab

ba

aa bbavaw bv bw

Figure 3.3(b). The configuration of (1, 0)Λ(E, F, p, q).

The complete and associative collection of commuting squares of Λ(E, F , p, q) is

hlC

aa

ggg

C k

bb

l klk

C h

aa

gkk

C

aa

aa

k

bb

bb

bb

l

lkC

ab

ggl

C khg

CkhC ll

ab

h

ab

ab

h g

ba

ba

ba

ba

k

Figure 3.3(c). Complete and associative collection of commuting squares of

Λ(E, F , p, q)

Observe that the skeletons and commuting squares of (E, F , p, q) and (1, 0)Λ(E, F,

p, q) are identical. So Λ(E, F , p, q) ∼= (1, 0)Λ(E, F, p, q)(see Corollary 3.50). We next

show that this is a general phenomenon. But first we need a technical result.

Proposition 3.45: Let (E, F, p, q) be a textile system such that p, q have unique path

lifting. For m ∈ N, the map ϕm : Fm → Λ(m,1)(E, F, p, q) given by ϕm(β1 . . . βm) =

q(β1) . . . q(βm)s(βm) is a bijection.

Proof. The map ϕm clearly maps Fm to Λ(m,1)(E, F, p, q). To see that it is injective,

suppose that ϕm(α) = ϕm(β). Then s(α) = s(β) and q(α) = q(β). Since q has

unique s-path lifting, this implies that α = β. To see that ϕm is surjective, fix

λ ∈ Λ(m,1)(E, F, p, q), factorize λ = µ1µ2 . . . µmν where each d(µi) = (1, 0) and d(ν) =

(0, 1). Then each µi ∈ E1 and v ∈ F 0. Since s(µm) = q(v), unique path lifting for

q implies that there exists β ∈ Fm such that q(β) = µ1µ2 . . . µm and s(β) = q(v).

Hence ϕm(β) = λ. 2


Remark 3.46: The map ϕm : Fm → Λ(m,1)(E, F, p, q) can be equivalently expressed

as ϕm(β1 . . . βm) = q(β1) . . . q(βm)s(βm) = r(β1)p(β1) . . . p(βm).

Definition 3.47: For m ∈ N and n ∈ N\0, given a textile system (E, F, p, q), de-

fine a directed graph E(m,m+n) = (Em, Em+n, r, s) by r(e1e2 . . . em+n) = e1e2 . . . em

and s(e1e2 . . . em+n) = en+1en+2 . . . em+n, and define F (m,n) similarly.

When m = 1, n = 1, we have E(1, 2) = E. We write Ea(m,m + n) for the set

(E(m,m + n))a of paths of length a in the graph E(m,m + n).

Theorem 3.48: Let (E, F, p, q) be a textile system such that p, q have path lifting,

and let m,n ∈ N. Then

(1). (E(m,m+n), F (m,m+n), p, q) is a textile system such that p, q have path

lifting; and

(2). If p, q have unique path lifting, so do p and q, and then if mn∈ N, we have

Λ(E(m,m + n), F (m,m + n), p, q) ∼= (mn, 0)Λ(E(0, n), F (0, n), p, q).

Proof. (1). We check that p, q are graph morphisms. Fix e1 . . . em+n ∈ E1(m,m+n).

Then

p(r(e1 . . . em+n) = p(e1 . . . em) = p(e1) . . . p(em)

= r(p(e1) . . . p(em+n)) = r(p(e1 . . . em+n),

and

p(s(e1 . . . em+n) = p(en+1 . . . em+n) = p(en+1) . . . p(em+n)

= s(p(e1) . . . p(em+n)) = s(p(e1 . . . em+n).

So p is a graph morphism. Similar calculations show that q is a graph morphism.

We show that p, q are surjective. Fix e1 . . . em+n ∈ E1(m,m + n). Since p is sur-

jective, there exists f1 ∈ F 1 such that p(f1) = e1. Since p has path lifting, Lemma 3.2

implies that there exists f2 . . . fm+n ∈ Fm+n−1 such that s(f1) = r(f2 . . . fm+n) and

p(f2 . . . fm+n) = e2 . . . em+n. Hence p(f1 . . . fm+n) = p(f1) . . . p(fm+n) = e1 . . . em+n.

Thus p is surjective.


To see that p has r-path lifting, fix e1 . . . em+n ∈ E1(m,m + n) and f1 . . . fm ∈F 0(m,m + n) such that p(f1 . . . fm) = r(e1 . . . em+n) = e1 . . . em ∈ E0(m,m + n).

Then r(em+1) = p(s(fm)), so that p has r-path lifting implies that there exist

fm+1 . . . fm+n ∈ F n such that p(fm+1 . . . fm+n) = p(fm+1) . . . p(fm+n) = em+1 . . . em+n

and r(fm+1 . . . fm+n) = r(fm+1) = s(fm) = s(f1 . . . fm). Hence for any e1 . . . em+n ∈E1(m,m+n) and f1 . . . fm ∈ F 0(m,m+n) such that p(f1 . . . fm) = e1 . . . em, there ex-

ist f1 . . . fm+n ∈ F 1(m,m + n) with p(f1 . . . fm+n) = e1 . . . em+n and r(f1 . . . fm+n) =

f1 . . . fm. Thus p has r-path lifting. A similar argument shows that p has s-path

lifting. The proof that q is surjective and has r-path lifting and s-path lifting is

similar.

(2). Next we want to show that if p, q have unique path lifting, p, q also have

unique path lifting. Fix e1 . . . em+n ∈ E1(m,m + n) and fix f1 . . . fm ∈ F 0(m,m + n)

such that p(f1 . . . fm) = r(e1 . . . em+n) = e1 . . . em ∈ E0(m,m + n). Suppose that

α, β ∈ F 1(m,m + n) satisfy that r(α) = r(β) = f1 . . . fm and p(α) = p(β) =

e1 . . . em+n. Then α = f1 . . . fmg1 . . . gn and β = f1 . . . fmh1 . . . hn and we have

r(g1) = r(h1) = s(fm), p(g1 . . . gn) = p(g1) . . . p(gn) = e1 . . . en and p(h1 . . . hn) =

p(h1) . . . p(hn) = e1 . . . en. Since p has unique r-path lifting, Lemma 3.2 forces

h1 . . . hn = g1 . . . gn, so α = β. Hence p has unique r-path lifting. The symmetric

argument using that p has unique s-path lifting instead unique r-path lifting shows

that p has unique s-path lifting. A similar argument shows that q has unique path

lifting.

Now fix m,n ∈ N satisfying mn∈ N. We have that

Λ0(E(m,m + n), F (m,m + n), p, q)

= E0(m,m + n) = Em = Emn (0, n)

= Λ(mn

,0)(E(0, n), F (0, n), p, q)

= ((m

n, 0)(Λ(E(0, n), F (0, n), p, q)))0. (3.7)

Also

Λ(1,0)(E(m,m + n),F (m,m + n), p, q)

= E1(m,m + n) = Em+n = Emn

+1(0, n)

= Λ(mn

+1,0)(E(0, n), F (0, n), p, q)


= ((m

n, 0)Λ(E(0, n), F (0, n), p, q)))(1,0), (3.8)

and Proposition 3.45 gives a bijection

ϕm : Fmn (0, n) → Λ(m

n,1)(E(0, n), F (0, n), p, q). We have

Fmn (0, n) = Λ(0,1)(E(m,m + n), F (m,m + n), p, q) and

Λ(mn

,1)(E(0, n), F (0, n), p, q) = ((m

n, 0)(Λ(E(0, n), F (0, n), p, q))(0,1).

So ϕm gives a bijection from Λ(0,1)(E(m,m+n), F (m,m+n), p, q) to ((mn, 0)(Λ(E(0, n),

F (0, n), p, q))(0,1).

We will write EblueΛ (m,m + n) for the edges α ∈ E1

Λ(m,m + n) with c(α) = 1

and EredΛ (m,m + n) for the edges β ∈ E1

Λ(m,m + n) with c(β) = 2. For (E(m,m +

n), F (m,m + n), p, q), define a 2-coloured graph EΛ(m,m + n) and a complete and

associative collection of squares CΛ(m,m+n) in EΛ(m,m + n) as in Definition 2.32.

Similarly, define a 2-coloured graph E(mn

,0)Λ and a complete and associative collection

of squares C(mn

,0)Λ in E(mn

,0)Λ as in Definition 2.32.

We claim that there is a coloured graph morphism ψ : EΛ(m,m + n) → E(mn

,0)Λ.

By (3.7) and (3.8), we have E0(m

n,0)Λ = v : v ∈ E0

Λ(m,m + n) and Eblue(m

n,0)Λ =

f : f ∈ EblueΛ (m,m + n). By definition Ered

(mn

,0)Λ = f : f ∈ Λ(0,1)(E(m,m +

n), F (m,m + n), p, q) and Ered(m

n,0)Λ = f : f ∈ ((m

n, 0)(Λ(E(0, n), F (0, n), p, q))(0,1).

So the bijection ϕm above determines a bijection ψred(f) = ϕm(f) from EredΛ (m,m+n)

to Ered(m

n,0)Λ. Then define a coloured graph morphism ψ : EΛ(m,m + n) → E(m

n,0)Λ as

follows

ψ0(v) = v for v ∈ E0Λ(m,m + n), (3.9)

ψ1(λ) = λ for λ ∈ EblueΛ (m,m + n), and (3.10)

ψ1(µ) = ψred(µ) for µ ∈ EredΛ (m,m + n). (3.11)

We check that ψ is a graph morphism. Fix λ ∈ EblueΛ (m,m + n) and µ = f ∈

EredΛ (m,m + n). Then r(ψ1(λ)) = r(λ) = ψ0(r(λ)); and r(ψ1(µ)) = r(ϕm(f)) =

ψ0(r(f)) and similarly at the source, so ψ is a graph morphism. By definition of ψ,

ψ preserves colour. So ψ is a coloured-graph isomorphism as claimed.

To see that ψ(CΛ(m,m+n)) = C(mn

,0)Λ, it is suffices to show that whenever e, e′ ∈Λ(1,0)(E(m,m+n), F (m,m+n), p, q) and f, f ′ ∈ Λ(0,1)(E(m,m+n), F (m,m+n), p, q)


satisfy ef = f ′e′, we have ψ(e)ψ(f) = ψ(f ′)ψ(e′) in (mn, 0)(Λ(E(0, n), F (0, n), p, q).

Let λ = ef = f ′e′. Then the commuting squares λ for some µ ∈ Fm+n are illustrated

as follows:

( , )n n m

ˆ ( )p

(0, )m

ˆ( )q

Figure 3.4. The commuting squares for some µ ∈ Fm+n.

So e = µ(0,m), f = p(µ), e′ = µ(n,m + n) and f ′ = q(µ). We have Fm+n =

Fmn (0, n) and Em+n = E

mn (0, n) as sets, and these identifications intertwine the maps

p, q : Fm+n → Em+n with p, q : Fmn (0, n) → E

mn (0, n). Moreover, when µ is viewed

as an element of ((mn, 0)Λ(E(0, n), F (0, n), p, q))(1,1), µ(0, (0, 1)) is equal to r(µ) cal-

culated in (mn)(F (0, n)) = F (m,m+n), which is µ(0,m). Similarly, µ((1, 0), (1, 1)) =

µ(n, n+m). We also have ψ(e) = ψ(µ(0,m)) = ϕm(µ(0,m)) = µ(0,m), and similarly

for the other edges, so ψ(e)ψ(f) = ψ(f ′)ψ(e′) as claimed.

Hence ψ : EΛ(m,m + n) → E(mn

,0)Λ is a coloured graph isomorphism such that

ψ φ ∈ CΛ(m,m+n) for all φ ∈ C(mn

,0)Λ. [18, Theorem 4.5] implies that Λ(E(m,m +

n), F (m,m + n), p, q) ∼= (mn, 0)Λ(E(0, n), F (0, n), p, q). 2

Corollary 3.49: Let (E, F, p, q) be a textile system such that p, q have path lifting.

For m ∈ N(1) (E(m,m + 1), F (m,m + 1), p, q) is a textile system such that p, q have path

lifting; and

(2) If p, q have unique path lifting, so do p and q, and then Λ(E(m,m +

1), F (m,m + 1), p, q) ∼= (m, 0)Λ(E(0, 1), F (0, 1), p, q).

Proof. (1). Recall Theorem 3.48, (E(m,m+1), F (m,m+1), p, q) is the special case

of when n = 1 in (E(m,m + n), F (m,m + n), p, q). Since (E(m,m + n), F (m,m +

n), p, q) is a textile system, then (E(m,m + 1), F (m,m + 1), p, q) is a textile system.

An argument like that of Theorem 3.48 shows that p, q have path lifting.


(2). With similar calculation as in Theorem 3.48, we have that if p, q have unique

path lifting, so do p and q. Moreover,

Λ0(E(m,m + 1), F (m,m + 1), p, q) = ((m, 0)Λ(E(0, 1), F (0, 1), p, q))0;

Λ(1,0)(E(m,m + 1), F (m,m + 1), p, q) = ((m, 0)Λ(E(0, 1), F (0, 1), p, q))(1,0);

and Proposition 3.45 gives a bijection ϕm : Fm → (m, 0)Λ(0,1)(E(0, 1), F (0, 1), p, q).

So the argument of Theorem 3.48 gives a coloured graph morphism ψ from EredΛ (m,m+

1) to Ered(m,0)Λ and we have

Λ(E(m,m + 1), F (m,m + 1), p, q) ∼= (m, 0)Λ(E(0, 1), F (0, 1), p, q). 2

More specially, when m = 1, n = 1, we have the conclusion that:

Corollary 3.50: Let (E, F, p, q) be a textile system such that p, q have path lifting.

Then

(1) (E, F , p, q) is a textile system such that p, q have path lifting; and

(2) If p, q have path lifting, so do p and q, and then (E, F , p, q) ∼= (1, 0)Λ(E, F, p, q).

4. TEXTILE SYSTEMS AND TILINGS

In [43], Nasu introduces LR resolving conditions on the graph homomorphisms p,

q to establish a connection between two-dimensional shifts of finite type and textile

systems. In this chapter, in order to construct tilings from textile systems, we first

show how path lifting extends to infinite paths.

Lemma 4.1: Let (E, F, p, q) be a textile system, and let (T, K, n, e, s, w) be the

associated system of Wang tiles. Suppose p, q have r-path lifting and s-path lift-

ing. Let · · · f−2f−1f0f1f2 · · · be an infinite path in F . Then there exist infinite

paths · · · g−2g−1g0g1g2 · · · and · · ·h−2h−1h0h1h2 · · · in F such that s(Tgi) = n(Tfi

)

and n(Thi) = s(Tfi

) for all i.

Proof. We will inductively choose edges gi ∈ F 1 such that q(gi) = ei and s(gi) =

r(gi+1) for all i. Since q is surjective, there exists g0 such that q(g0) = e0. Now

suppose we have chosen g−n . . . g−1g0g1 . . . gn such that q(gi) = ei and r(gi+1) = s(gi)

for all i. We have q(s(gn)) = s(q(gn)) = s(en) = r(en+1), since q has r-path lifting,

there exists gn+1 ∈ F 1 such that q(gn+1) = en+1 and r(gn+1) = s(gn).

Similarly, since q has s-path lifting, there exists g−n−1 ∈ F 1 such that q(g−n−1) =

e−n−1 and s(g−n−1) = r(g−n). Now by induction there exist gi : i ∈ Z in F with

q(gi) = ei, q(r(gi)) = r(ei) and q(s(gi)) = s(ei). So the infinite path · · · g−2g−1g0g1g2 · · ·satisfies s(Tgi

) = n(Tfi) for all i. Since p has r-path lifting and s-path lifting, sim-

ilarly to the above, there exists an infinite path · · ·h−2h−1h0h1h2 · · · in XF with

n(Thi) = s(Tfi

) for all i. 2

4. Textile systems and tilings 56

2f

1f

0f

1f 2

f

F

E

( )i

p f( )i

q g

2e

1e

0e

1e

2e

2g 1

g0

g 1g

2g

Figure 4.1. The configuration in Lemma 4.1.

Notation 4.2: For α ∈ XF , we write p−1q(α) for the set β ∈ XF : p(β) = q(α),and q−1p(α) = γ ∈ XF : p(α) = q(γ).

The hypothesis that p, q have path lifting in Lemma 4.1 is necessary as the fol-

lowing example demonstrates.

Example 4.3: Define E0 = w, E1 = e1, e2, F 0 = u, v, F 1 = f1, f2. Define

rF , sF : F 1 → F 0 and rE, sE : E1 → E0 by r(f1) = s(f1) = u, r(f2) = u and s(f2) = v;

r(e1) = s(e1) = r(e2) = s(e2) = w. Define the morphisms p, q as p(f1) = e1, p(f2) = e2

and q(f1) = e2, q(f2) = e1. Then p(v) = q(v) = w = r(e2), but there is no edge whose

range is v, therefore p and q both fail to have r-path lifting.

:F

:E

v

2e

1e

u

w

1f

2f

Figure 4.2. A textile system in which p, q do not have r-path lifting.

Let α = . . . f1f1f1 . . . ∈ XF . Then p(α) = . . . e1e1e1 . . . ∈ XE. We have q−1(e1) =

f2, but r(f2) 6= s(f2), so implies that q−1 p(α) = ∅.

In Example 2.11 and Example 4.3, we have written out the formal definition of

the graphs involved as well as drawing visual representations. Henceforth, we will


usually just draw the pictures since they contain all of the necessary information and

are easier to read.

Lemma 4.4: Let (E, F, p, q) be a textile system. Suppose that |E0| = |F 0| < ∞,

then p and q have r-path lifting and s-path lifting.

Proof. Since |E0| = |F 0| < ∞ and since p, q are surjective, p and q are also

injective on F 0. Let e ∈ E1 and w ∈ F 0 satisfy r(e) = p(w). Since p is surjective,

there exists f ∈ F 1 such that p(f) = e. Since p is a graph morphism, p(w) = r(e) =

r(p(f)) = p(r(f)). Since p is injective, it follows that w = r(f). Hence p has r-path

lifting. The same reason shows that when s(e) = p(v), there exists g ∈ F 1 such

that p(v) = s(e) = s(p(g)) = p(s(g)) and v = s(g). Hence p has s-path lifting. A

symmetric argument shows that q has r-path lifting and s-path lifting. 2

Lemma 4.5: Let (E, F, p, q) be a textile system. Suppose that |E1| = 1. For any

pair of infinite paths · · · g−2g−1g0g1g2 · · · and · · ·h−2h−1h0h1h2 · · · in F , we have

s(Tgi) = n(Thi

) for all i. If F 2 is nonempty, then the unique e ∈ E1 is a loop.

Proof. Since E1 = e, s(Tgi) = q(gi) = p(gi) = p(hi) = n(Thi

). Now suppose

fg ∈ F 2. Then s(e) = s(p(f)) = p(s(f)) = p(r(g)) = r(p(g)) = r(e). Hence e is a

loop. 2

To use path lifting for a textile system (E, F, p, q) in order to get a valid tiling,

we use the following proposition.

Proposition 4.6: Let (E, F, p, q) be a textile system in which p, q have r-path lifting

and s-path lifting. Let (T, K, n, e, s, w) be the associated system of Wang tiles. Then

for each bi-infinite path · · · f−2f−1f0f1f2 · · · in F there exists a tiling x of Z2 such

that x(i, 0) = Tfifor all i ∈ Z.

Proof. Let f 0 := · · · f−2f−1f0f1f2 · · · . Since p, q have r-path lifting and s-path

lifting, Lemma 4.1 gives bi-infinite paths g and h in F such that s(Tgi) = n(Tfi

) and

n(Thi) = s(Tfi

) for all i.

Inductively, for all i > 0, given the bi-infinite paths · · · f i−2f

i−1f

i0f

i1f

i2 · · · and

· · · f−i−2f

−i−1f

−i0 f−i

1 f−i2 · · · , Lemma 4.1 gives bi-infinite paths · · · f i+1

−2 f i+1−1 f i+1

0 f i+11 f i+1

2 · · ·


and · · · f−i−1−2 f−i−1

−1 f−i−10 f−i−1

1 f−i−12 · · · such that s(Tf i+1

j) = n(Tf i

j) and s(Tf−i+1

j) =

n(Tf−ij

) for all j ∈ Z. So, x(m,n) := Tfnm

is the desired tiling. (See Figure.8) 2

2f

1f

0f

1f

2f

F

E

( )i

p f1

( )i

q f

2e

1e 0

e1

e 2e

1

2f

1

2e

1( )

ip f

2( )

iq f

1

1f

1

0f

1

1f

1

2f

2

2f

2

1f

2

0f

2

1f

2

2f

1

1e

1

0e

1

1e

1

2e

1

2f 1

1f

1

0f

1

1f

1

2f

( )i

q f1

( )i

p f

1

2e

1

1e

1

0e

1

1e

1

2e

Figure 4.3. The textile system of (E, F, p, q).

2fT

1fT

0fT

1fT

2fT

2e 1

e0

e1

e2

e

12f

T

1

2e

11f

T 10f

T 11f

T 12f

T

22f

T 21f

T 20f

T 21f

T 22f

T

12f

T 11f

T 10f

T1

1fT 1

2fT

1

1e

1

0e

1

1e

1

2e

1

2e

1

1e

1

0e

1

1e

1

2e

Figure 4.4. The tiling x: Z2 → T (in this diagram, the shaded square Tf0 indicates

where the origin x(0, 0) is).

We give an example of Proposition 4.6 in action. Recall the textile system of

Example 2.11. Then p, q have r-path lifting and s-path lifting. Let P0 be the infinite

path . . . lkhglkhg . . . in F . Then p(P0) = . . . babababa . . .. Using q path lifting, we

find another infinite path P1 = . . . hglkhglk . . ., with q(P1) = p(P0) = . . . babababa . . ..


Similarly, since p(P1) = . . . babababa . . ., we will find an infinite path P2 ∈ P0, P1,with q(P2) = p(P1). Now by induction, we can expand the associated Wang tiles Tfi

to the whole upper half-plane. Similarly, when we consider q(P0) = . . . babababa . . .,

we can expand the associated Wang tiles Tfito the whole lower half-plane. (See

Figure. 4.5) The shaded tile Tk is denoted as x(0, 0).

b a

kT

hT

gT

lT

hT

gT

lT

kT

b

b

b

b

b

b

ba

a

a

a

a

a

a

hT

hT

hT

hT

lT

lT

lT

lT

gT

gT

gT

gT

kT

kT

kT

kT

b b b b

b b b b

a

a

a

a

a

a

a

a

Figure 4.5. The tiling x of (T, K, n, e, s, w).

Lemma 4.7: Let (E, F, p, q) be a textile system in which p, q have r-path lifting and

s-path lifting. Let (T, K, n, e, s, w) be the associated system of Wang tiles. If XE 6= ∅then there exists a tiling x of Z2 by T .

Proof. Fix an infinite path · · · e−2e−1e0e1e2 · · · in E. Since p, q are onto and have

r-path lifting and s-path lifting, Lemma 4.1 implies that there exists an infinite path

· · · f−2f−1f0f1f2 · · · in F such that q(· · · f−2f−1f0f1f2 · · · ) = · · · e−2e−1e0e1e2 · · · .Proposition 4.6 therefore implies that there exists a tiling x of Z2 by T . 2

Corollary 4.8: Let (E, F, p, q) be a textile system with no sinks and sources in which

p, q are onto and have r-path lifting and s-path lifting. Let (T, K, n, e, s, w) be the

associated system of Wang tiles. Let m ∈ N2 and suppose that b : [0,m] → T satisfies

n(b(i, j)) = s(b(i, j + 1)) and e(b(i, j)) = w(b(i + 1, j)) for all i, j. Then there exists

a tiling x by T such that x|[0,m] = b.

Proof. Fix a sequence of paths γ0, γ1, . . . , γm2 ∈ Fm1 such that Tγj(i) = b(i, j) for all

i, j ∈ [0,m]. Since F has no sources, there exists an one-sided infinite path α0 ∈ X+F


such that r(α0) = s(γ0). Since p, q have r-path lifting, Lemma 4.1 implies that there

exists a one-sided infinite path α1 ∈ X+F such that r(α1) = s(γ1) and q(α1) = p(α0).

Inductively, we can find a sequence of one-sided infinite paths αk ∈ X+F such that

r(αk) = s(γk) for all 2 ≤ k ≤ m2 and q(αk+1) = p(αk) for all 2 ≤ k ≤ m2 − 1.

Similarly, since F has no sinks, and since p, q are onto and have s-path lifting,

Lemma 4.1 implies that there is a sequence of one-sided infinite paths βk ∈ X−F

such that s(βk) = r(γk) for all 0 ≤ k ≤ m2 and q(βk+1) = p(βk) for all 0 ≤k ≤ m2 − 1. Hence, λi = βiγiαi are bi-infinite paths in F ∗ such that p(βiγiαi) =

q(βi+1γi+1αi+1). Now define x+ : [(−∞, 0], (+∞,m2]] → T as follows: for (n1, n2) ∈[(−∞, 0], (+∞,m2]], define x+(n1, n2) = Tλn2 (n1). Then x+|[0,m] = b. Since p, q have

path lifting, and λ0, λm2 are bi-infinite paths in XF , we can use Lemma 4.1 repeatedly

as in the proof of Proposition 4.6 to see that there exists a tiling x : Z2 → T satisfying

n− s and e−w conditions and x|[(−∞,0],(+∞,m2]] = x+. Hence x|[0,m] = x+|[0,m] = b. 2

Next, we will introduce the essential subsystem (Eess, Fess, p, q) of a textile system

(E, F, p, q) and show that they have identical shift spaces.

Let E be directed graph and suppose that F 0 ⊆ E0 and F 1 ⊆ E1 satisfy

rE(f), sE(f) ∈ F 0 whenever f ∈ F 1. Then the quadruple F = (F 0, F 1, r|1F , s|1F )

is a directed graph, which we call a subgraph of E. We usually just write r, s for the

range and source maps on F 1.

Definition 4.9: A sink (or ‘dead end’) is a vertex which emits no edges, so s−1(v) =

∅. A source is a vertex which receives no edges, so r−1(v) = ∅.

It is obvious that sinks and sources cannot occur in any bi-infinite walk in a

directed graph E. Our results allow for graphs with sinks and sources but infinite

paths and tilings never involve edges from which one can only reach sources or can

only be reached from sinks.

Definition 4.10: Let E be a directed graph. A vertex v ∈ E0 is stranded if the

vertex is a sink or a source (or both).

To remove stranded vertices, we now detail how to replace a textile system

(E, F, p, q) with another (Eess, Fess, p, q) which has the same collection of tilings and

in which Eess, Fess have no sinks or sources.


Definition 4.11: A directed graph E is called essential if no vertex of E is stranded.

An essential subgraph F of E is maximal if whenever we have another essential

subgraph G of E with F ⊆ G, then F = G.

Lemma 4.12: Let E be a directed graph, and F1 and F2 be maximal essential sub-

graphs of E. Then F1 = F2.

Proof. Firstly, we define (F1 ∪F2)0 = F 0

1 ∪F 02 and (F1 ∪F2)

1 = F 11 ∪F 1

2 . Since F1

and F2 are essential subgraphs of E, the graph F1 ∪ F2 is also an essential subgraph

of E: for if not, then there must be vertices stranded in F1 ∪ F2, and the vertices

must be stranded in F1 or F2 (or both), which contradicts the assumption. Since

F1 ⊆ F1 ∪F2 that F1 is a maximal essential subgraph implies that F1 = F1 ∪F2, thus

F2 ⊆ F1. By symmetry, we also have F1 ⊆ F2. Hence, F1 = F2. 2

Definition 4.13: Let E be a directed graph. The maximal essential subgraph of a

graph E is denoted as Eess.

Remark 4.14: If E has no cycles, then Eess is empty.

Definition 4.15: We say a directed graph E is locally finite if every vertex in E0

has finite degree, such that for any v ∈ E0, |r−1(v)| < ∞ and |s−1(v)| < ∞.

Note that any finite graph is locally finite; however, infinite graphs can also be

locally finite.

Lemma 4.16: Let E be a locally finite graph. If v receives a path of length n for

every n ∈ N, then v receives an infinite path in E. If v ∈ E0 emits a path of length

n for every n ∈ N, then v emits an infinite path in E.

Proof. Let (αn)∞n=1 be a sequence of paths so that r(αn) = v for all n, and

|αn| = n for all n, so αn = αn1αn

2 . . . αnn. Since |s−1(v)| < ∞, there exists e1 ∈ E1

such that r(e1) = v and e1 = αn1 for infinitely many n. Let N1 = n ∈ N : αn

1 = e1,so |N1| is infinite. Since r−1(s(e1)) is finite, there exists e2 ∈ r−1(s(e1)) such that

N2 = n ∈ N1 : αn2 = e2 is infinite. Continuing this way, we obtain sets Ni ∈ N


and edges ei such that each Ni ⊆ Ni−1, |Ni| is infinite and for all n ∈ Ni, αni = ei.

Now e1e2e3 . . . is an infinite path received by v. Similarly, using the condition of

|s−1(v)| < ∞, we can get the infinite path . . . e−2e−1e0 emitted by v. 2

Lemma 4.17: Let E be a locally finite directed graph, and define:

(1) F 0 = v ∈ E0 : for every n ∈ N, there exist α, β ∈ En such that s(α) = v

and r(β) = v; and

(2) F 1 = e ∈ E1 : r(e), s(e) ∈ F 0.

Then F is a maximal essential subgraph of E, and hence F = Eess.

Proof. By definition, r(f), s(f) ∈ F 0 for all f ∈ F 1, and so F is a subgraph

of E. We want to prove that F is essential. Fix v ∈ F 0. By condition (1), v

emits a path of length n for every n and receives a path of length n for every n.

By Lemma 4.16, v receives an infinite path α1α2α3 . . . and emits an infinite path

. . . β3β2β1. We claim that s(α1) ∈ F 0. Fix n ∈ N. We must show that there exists

µ, ν ∈ En such that s(µ) = s(α1) and r(ν) = s(α1). The paths µ = βn−1 . . . β1α1 and

ν = α2α3 . . . αn+1 have the desired properties. Similarly, r(β1) ∈ F 0 because, for every

n ∈ N, ν = β1α1 . . . αn−1 ∈ En with r(ν) = r(β1), and µ = βn+1βn . . . β2 ∈ En with

s(µ) = r(β1). Since r(β1), s(β1) ∈ F 0, we have β1 ∈ F 1, and since s(α1), r(α1) ∈ F 0,

we have α1 ∈ F 1. In particular, v is not stranded in F . Hence F is essential.

Now, we want to show F = Eess. If not, F is not the maximal essential subgraph

of E. Then F ⊂ G := Eess. There are two cases to consider:

(i) There exists v ∈ G0 such that v /∈ F 0;

(ii) F 0 = G0 and there exists e ∈ G1 such that e /∈ F 1.

In (i), suppose that v ∈ G0, but v /∈ F 0. By (1), there exists n ∈ N such that

either v does not receive a path of length n in E or v does not emit a path of length n

in E. We consider the case that v does not emit a path of length n for some n. Then

there is a largest n ∈ N such that v emits a path of length n in E. Since G ⊆ E, v

does not emit a path of length n + 1 in G. This contradicts the assumption that G is

essential. The case that v does not receive a path of length n for some n is similar.

Hence F 0 = G0.


In (ii), we have F 0 = G0. By (2), F 1 = e ∈ E1 : r(e), s(e) ∈ F 0. Since

e /∈ F 1, we have either r(e) /∈ F 0 or s(e) /∈ F 0 (or both). But since F 0 = G0, this is

impossible. Thus, F 1 = G1. 2

Lemma 4.18: 1). Let E be a locally finite directed graph. Then the bi-infinite paths

of E are precisely the bi-infinite paths of Eess.

2). Let E, F be locally finite directed graphs. If (E, F, p, q) is a textile system

with r-path lifting and s-path lifting, then so is (Eess, Fess, p, q).

3). Let E, F be locally finite directed graphs. The tilings arising from (E, F, p, q)

are the same as the tilings arising from (Eess, Fess, p, q).

Proof. 1). Since Eess is a subgraph of E, all infinite paths in Eess are infinite paths

in E. Let α = . . . α−2α−1α0α1α2 . . . be a bi-infinite path in E. For each i, n ∈ N,

µ = αi−nαi−n+1 . . . αi−1 ∈ En with s(µ) = r(αi) and ν = αiαi+1 . . . αi+n−1 ∈ En with

r(ν) = r(αi). Hence Lemma 4.17 implies r(αi) ∈ E0ess for all i. Since s(αi) = r(αi+1),

this gives r(αi), s(αi) ∈ E0ess for all i, and then Lemma 4.17 implies αi ∈ E1

ess for all

i. So α is an infinite path in Eess.

2). Firstly we want to prove (p(F 0ess), p(F 1

ess), r, s) and (q(F 0ess), q(F

1ess), r, s) are

essential. Fix v ∈ p(F 0ess). Then v = p(w) for some e ∈ F 0

ess. Since Fess is essential,

there exist e, f ∈ F 1ess such that r(e) = w = s(f). Hence p(e), p(f) ∈ p(F 1

ess)

satisfy r(p(e)) = p(r(e)) = v = p(s(f)) = s(p(f)). So v is not stranded. Hence

(p(F 0ess), p(F 1

ess), r, s) is essential. A similar argument shows that (q(F 0ess), q(F

1ess), r, s)

is essential.

We prove that p(Fess) = Eess and q(Fess) = Eess. To see that p(Fess) = Eess, fix

v ∈ E0ess. By part 1, for any n ∈ N, there exist α, β ∈ En such that r(α) = v and

s(β) = v. Since p has path lifting, there exist µ, ν ∈ F n such that p(r(µ)) = p(s(ν)) =

v and p(µ) = α and p(ν) = β. Applying Lemma 4.17 (1), we have r(α) = s(β) ∈ F 0ess.

Hence v ∈ p(F 0ess). Now suppose that e ∈ E1

ess. Choose a path α ∈ En such

that e = α0. Then by what we have proved above, each r(µi) ∈ F 0ess. Applying

Lemma 4.17 (2), we have µ0 ∈ F 1ess and e = p(µ0) ∈ p(F 1

ess). Thus p(Fess) = Eess.

With same argument, we have q(Fess) = Eess.


Finally, we want to prove p and q are graph morphisms of Eess and Fess with

p, q: Fess → Eess and that p|Fess and q|Fess have r-path lifting and s-path lifting.

By part 1, any v ∈ E0ess emits an infinite path in Eess and receives an infinite path

in Eess. Suppose the infinite path in Eess emitted by v is . . . ei−3ei−2ei−1 and the

infinite path in Eess received by v is eiei+1ei+2 . . ., by part 1, the bi-infinite paths

of Eess and Fess are precisely the bi-infinite paths of E and F . Since (E, F, p, q)

has r-path lifting and s-path lifting, Lemma 4.1 implies that there exist bi-infinite

paths α = . . . fi−2fi−1fifi+1fi+2 . . . and β = . . . gi−2gi−1gigi+1gi+2 . . . in F such that

p(fi) = ei and q(gi) = ei for all i. Also, by part 1, the bi-infinite pathes α, β ∈ F∞ess.

Thus, for every edge e ∈ E1ess, there exist edges f, g ∈ F 1

ess with p(f) = e, q(g) = e.

Hence, (Eess, Fess, p, q) is a textile system and has r-path lifting and s-path lifting.

3). If x is a tiling, then each row of x is in F , so in Fess, so the image of each

row of x belongs to Eess. Thus, the tilings arising from (E, F, p, q) are precisely the

tilings arising from (Eess, Fess, p, q). 2

We have shown that when considering the tilings associated to a textile system

(E, F, p, q), we may assume that E, F are essential.

We show next how to construct examples of textile systems using the skew-

product construction for directed graphs. Given a labeling c of the edges of a directed

graph E by elements of a discrete group G, we can form a skew-product graph E×cG.

We construct a textile system (E ×c G,E, π, α π) from the data E, c,G and an au-

tomorphism α of E. Before we talk about the textile system (E×c G,E, π, α π), we

give the definition of the skew-product graph E ×c G.

Definition 4.19: Let E be a locally finite directed graph. A labeling of E by a group

G is a function: c : E1 → G. The skew-product graph E ×c G is the directed graph

E×cG = ((E×cG)0, (E×cG)1, r, s) where (E×cG)0 = E0×G and (E×cG)1 = E1×G

with r(e, g) = (r(e), g) and s(e, g) = (s(e), gc(e)) for all (e, g) ∈ E1 ×G.

Since s−1(v, g) = (e, gc−1(e)) | e ∈ s−1(v), the vertex (v, g) ∈ (E ×c G)0 emits

the same number of edges as v ∈ E0. Similarly, since r−1(v, g) = (e, g) | r(e) = v,the vertex (v, g) ∈ (E ×c G)0 receives the same number of edges as v ∈ E0. Thus


E ×c G is locally finite if and only if E is locally finite, and (v, g) is a sink or source

if and only if v is a sink or source.

Remark 4.20: Define a map π : E ×c G → E by π0(v, g) = v, π1(e, g) = e. Then:

(1) (π0)−1(v) = (v, g) : g ∈ G; and

(2) (π1)−1(e) = (e, g) : g ∈ G.

Let E, F be two directed graphs. The Diamond Product graph [24] is defined to be

E¦F = (E0×F 0, E1×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)).

In [24], it is shown that if E is any directed graph, G = Z and c(e) = 1 for all e ∈ E1,

then E ×c G is identical to E ¦ F where F is the graph:

01

0 1 21

1 2

32

2

Example 4.21: Let E be the graph

v w

e

f

gh:E

and let G = Z. Define c : E1 → G by c(e) = c(f) = c(g) = c(h) = 1. Then E ×c G is

drawn as follows

( ,0)g ( ,1)g( , 1)g

( ,0)h( , 1)h ( ,1)h

( ,0)e ( ,0)f ( ,1)f( , 1)f ( ,1)e( , 1)e

( ,0)v ( ,1)v ( , 2)v( , 1)v

( ,0)w ( ,1)w ( , 2)w( , 1)w

Figure 4.6. Skew-product graph E ×c G.


Lemma 4.22: Let E be a directed graph, G a countable group, and c : E1 → G a

labeling. Define π : E ×c G → E by π0(v, g) = v, π1(e, g) = e, then

(i). π is a |G| to 1 surjective graph morphism;

(ii). π : E ×c G → E has unique r-path lifting and unique s-path lifting.

(iii). There is a bijection En ×G → (E ×c G)n given by

(e1, . . . en, g) → (e1, g)(e2, gc(e1))(e3, gc(e1)c(e2)) . . . (en, gc(e1)c(e2) . . . c(en−1)).

Proof. (i) The definitions of π0 and π1 give

π0(r(e, g)) = π0(r(e), g) = r(e) = r(π1(e, g)) and

π0(s(e, g)) = π0(s(e), gc(e)) = s(e) = s(π1(e, g)).

Thus, π is a |G| to 1 surjective graph morphism.

(ii) Fix v ∈ E0, e ∈ E1, s(e) = v and w ∈ (E ×c G)0 such that π(w) = v. By

Remark 4.20, w = (v, g) for some g ∈ G. Then (e, gc−1(e)) ∈ (E ×c G)1 satisfies

s(e, gc(e)−1) = (s(e), gc(e)−1c(e)) = (s(e), g) = (v, g) (4.1)

π1(e, gc(e)−1) = e. (4.2)

Therefore, π has s-path lifting. We claim that (e, gc−1(e)) is the only edge in

(E×cG)1 satisfying (4.1) and (4.2). To see this, suppose (f, h) is an edge in (E×cG)1,

such that s(f, h) = (v, g) and π1(f, h) = e. Since s(f, h) = (s(f), hc(f)) = (v, g), we

have s(f) = v and g = hc(f). Since π1(f, h) = e, we have π1(f, h) = f = e. Then

f = e and h = gc(e)−1.

Similarly, let u ∈ E0, m ∈ E1 with r(m) = u and n ∈ (E ×c G)0 such that

π(n) = u. Then by Remark 4.20, n = (u, g) for some g ∈ G. Then (m, g) ∈ (E×c G)1

satisfies

r(m, g) = (r(m), g) = (u, g) and π1(m, g) = m. (4.3)

Therefore, π has r-path lifting. We claim that (m, g) is the only edge in (E ×c G)1

satisfying (4.3). To see this, suppose (m′, g′) is an edge in (E ×c G)1, such that

r(m′, g′) = (u, g) and π1(m′, g′) = m. Since π1(m′, g′) = m′, then m′ = m, and since

r(m′, g′) = (r(m′), g′) = (r(m), g′) = (u, g′) = (u, g). So g′ = g.


(iii) Suppose that

(e1, g1)(e2, g2) . . . (en, gn) ∈ (E ×c G)n.

We claim that s(ei) = r(ei+1) and gi+1 = gic(ei) for all i. We prove this by induction.

When n = 2, the path (e1, g1)(e2, g2) ∈ (E ×c G)2 must have s(e1, g1) = r(e2, g2).

Since s(e1, g1) = (s(e1), g1c(e1)) and r(e2, g2) = (r(e2), g2), we have(s(e1), g1c(e1)) =

(s(e1), g1) = (r(e2), g2) = r(e2, g2). Hence, s(e1) = r(e2) and g2 = g1c(e1). Now

suppose as inductive hypothesis that s(ei) = r(ei+1) and gi+1 = gic(ei) for all 1 ≤i ≤ k − 1. Then s(ek−1) = r(ek) and gk = gk−1c(ek−1) = . . . = gc(e1)c(e2) . . . c(ek−1).

Since (e1, g1)(e2, g2) . . . (ek+1, gk+1) ∈ (E×cG)k+1, we have s(ek) = r(ek+1) and gk+1 =

gkc(ek) = gc(e1)c(e2) . . . c(ek). Thus, the claim holds.

Hence, the formula

(e1, . . . en, g) → (e1, g)(e2, gc(e1))(e3, gc(e1)c(e2)) . . . (en, gc(e1)c(e2) . . . c(en−1))

gives a bijection between (E ×c G)n and En × G. In particular, the gi are totally

specified by g1 and c(e1)c(e2) . . . c(en−1). 2

By Lemma 4.22, there is a bijection between X+E×cG and X+

E ×G given by

(x1, x2, . . . , g) 7→ (x1, g)(x2, gc(x1)) . . . .

Next we will inspect two-sided infinite paths in XE ×c G.

Lemma 4.23: Let E be a directed graph, let G be a discrete group and let c : E1 →G be a labelling. Then

(. . . x−2x−1x0x1x2 . . . , g) →

. . . (x−1, gc(x0)−1c(x−1)

−1)(x0, gc(x0)−1)(x1, g)(x2, gc(x1)) . . . is a bijection from

XE ×G to XE×cG.

Proof. Let . . . , x−1, x0, x1, . . . be consecutive edges in E. Then as above

(x1, g)(x2, gc(x1)) . . . (xi, gc(x1)c(x2) . . . c(xi−1)) . . .

is a one-sided infinite path in X+E×cG.


We can check as above that . . . (x−1, gc(x0)−1c(x−1)

−1)(x0, gc−1(x0)) is a one-sided

infinite path:

s(x−i, gc(x0)−1c(x−1)

−1 . . . c(x−i)−1)

= (s(x−i), gc(x0)−1c(x−1)

−1 . . . c(x−i)−1c(x−i))

= (s(x−i), gc(x0)−1c(x−1)

−1 . . . c(x−i+1)−1), and

r(x−i+1, gc(x0)−1c(x−1)

−1 . . . c(x−i+1)−1)

= (r(x−i+1), gc(x0)−1c(x−1)

−1 . . . c(x−i+1)−1).

Since . . . , x−1, x0 are consecutive edges in E, we have s(x−i) = r(x−i+1) for all

i = 1, 2, . . .. Hence,

s(x−i, gc(x0)−1c(x−1)

−1 . . . c(x−i)−1) = r(x−i+1, gc(x0)

−1c(x−1)−1 . . . c(x−i+1)

−1).

Since

r(x1, g) = (r(x1), g) = (s(x0), gc−1(x0)c(x0)) = s(x0, gc(x0)−1),

we have that (x0, gc(x0)−1) and (x1, g) are consecutive edges in E ×c G. Hence,

. . . (x−1, gc(x0)−1c(x−1)

−1)(x0, gc(x0)−1)(x1, g)(x2, gc(x1)) . . . ∈ XE×cG.

Conversely, suppose that

. . . (x−2, g−2)(x−1, g−1)(x0, g0)(x1, g1)(x2, g2) . . . ∈ XE×cG,

then each (r(xi), gi) = r(xi, gi) = s(xi−1, gi−1) = (s(xi−1), gi−1c(xi−1)). Hence, we

have . . . x−2x−1x0x1x2 . . . ∈ XE, and an induction gives gi = g1c(x1)c(x2) . . . c(xi) for

i ≥ 1 and gi = g1c(x0)−1c(x−1)

−1 . . . c(x−i)−1 for i < 1. 2

We now show how to use a combination of a labelling and an automorphism to

construct examples of textile systems.

Lemma 4.24: Let E be a graph. Let α be an automorphism of E and let G be a

group. Define π : E×c G → E by π0(v, g) = v, π1(e, g) = e. Then (E×c G,E, π, απ)

is a textile system, and π and α π have unique r-path lifting and s-path lifting.


Proof. Lemma 4.22 (i), (ii) implies that π : E×c G → E is a graph morphism with

unique path lifting. We must show that α π is a graph morphism from E ×c G to

E and that (e, g) → (π1(e, g), (α1 π1)(e, g), r(e, g), s(e, g)) is injective.

To see that α π is a graph morphism, fix e ∈ E1 and (e, g) ∈ (E ×c G)1. Since

(α π)0(r(e, g)) = (α0 π0)(r(e), g) = α0(r(e)) = r(α1(e))

= r((α1 π1)(e, g)) = r((α π)1(e, g)); and

(α π)0(s(e, g)) = (α0 π0)(s(e), gc(e)) = α0(s(e)) = s(α1(e))

= s((α1 π1)(e, g)) = s((α π)1(e, g)),

α π is a graph morphism.

To see that απ is surjective, fix v ∈ E0. Then π(v, g) = v for any g ∈ G, and there

exists (α−1(v), g) ∈ (E ×c G)0 such that (α π)0(α−1(v), g) = α0(π0(α−1(v), g)) = v

for any g ∈ G. Similarly, fix e ∈ E1. Then π(e, g) = e for any g ∈ G, and there exists

(α−1(e), g) ∈ (E ×c G)1 such that (α π)1(α−1(e), g) = α1(π1(α−1(e), g)) = e for any

g ∈ G.

Thus α π is a |G| to 1 surjective graph morphism. To see that (e, g) →(π1(e, g), (α1 π1)(e, g), r(e, g), s(e, g)) is injective, suppose that

(π1(e, g), (α1 π1)(e, g), r(e, g), s(e, g)) = (π1(f, h), (α1 π1)(f, h), r(f, h), s(f, h)).

Since π1(e, g) = e and π1(f, h) = f , we have e = f . Moreover, since

(r(e), g) = r(e, g) = r(e, h) = (r(e), h),

we have g = h. Therefore, (e, g) → (π1(e, g), (α1π1)(e, g), r(e, g), s(e, g)) is injective.

Hence, (E ×c G,E, π, α π) is a textile system.

Now, we show that α π has unique path lifting. Consider r-path lifting. Fix

e ∈ E1. Suppose that (w, g) ∈ (E ×c G)0 satisfies (α0 π0)(w, g) = r(e). Then

(α−1(e), g) satisfies r(α−1(e), g) = (w, g) and (α1 π1)(α−1(e), g) = e. Moreover, if

(f, h) ∈ (E ×c G)1 satisfies r(f, g) = (w, g) and (α1 π1)(f, h) = e, then (w, g) =

r(f, h) = (r(f), h), so h = g; and since e = (α1 π1)(f, h) = α1 π1(f, h) = α1(f), we

have f = (α1)−1(e).

A similar argument establishes unique s-path lifting. 2


Lemma 4.25: Let E be a graph. Let G be a group and let c : E1 → G be a

labelling. Define π : E ×c G → E by π0(v, g) = v, π1(e, g) = e. Suppose that

λ = e1e2 . . . en ∈ En satisfies s(en) = r(e1) and c(e1)c(e2) . . . c(en) ∈ G has finite

order: (c(e1)c(e2) . . . c(en))m = e for some m. For each v ∈ π−1(r(λ)), there is a

unique µ ∈ (E ×c G)mn such that π(µ) = λλ . . . λ︸︷︷︸m

and r(µ) = v, and then s(µ) = v

also.

Proof. From the definition of π, by Lemma 4.22.(i),(ii), π is a |G| to 1 graph

morphism, and π : E ×c G → E has unique r-path lifting and unique s-path lifting.

Thus from Lemma 4.1, there exists a unique path

λ′ = (e1, g1)(e2, g1c(e1)) . . . (en, g1c(e1) . . . c(en−1))

(e1, g1c(e1) . . . c(en))(e1, g1c(e1) . . . c(en)c(e1)) . . . (e1, g1(c(e1) . . . c(en))2c(en)−1)

. . .

(e1, g1(c(e1) . . . c(en))m−1) . . . (e1, g1(c(e1) . . . c(en))m)c(en)−1)

such that λ′ ∈ (E ×c G)nm with

r(λ′) = r(e1, g) = (r(e1), g) = r(µ) and π(λ′) = λm. (4.4)

By unique path lifting, λ′ is the only path satisfying (4.4). Hence

µ = (e1, g1) . . . (en, gn)(e1, gn+1) . . . (en, g2n) . . . (e1, g(m−1)n+1) . . . (en, gnm),

where

gnm = g1c(e1)c(e2) . . . c(en−1)c(en) . . . c(enm−1)

= g1(c(e1)c(e2) . . . c(en))m−1c(e1)c(e2) . . . c(en−1).

We have r(µ) = r(e1, g1) = (r(e1), g1), and

s(µ) = s(en, gnm)

= (s(en), gnmc(en))

= (s(en), g1(c(e1)c(e2) . . . c(en))m−1c(e1)c(e2) . . . c(en−1)c(en))

= (s(en), g1(c(e1)c(e2) . . . c(en))m).

Since c(e1)c(e2) . . . c(en) has order m in G, we have s(µ) = (s(en), g1), and hence

s(en) = r(e1), r(µ) = s(µ). 2


Now we discuss periodicity for textile systems. We identify conditions which guar-

antee that there exist periodic tilings consistent with a given textile system (E, F, p, q).

Firstly, we give the definition of shift maps and periodicity of a tiling.

Recall from Definition 2.12 that if (T, K, n, e, s, w) is a system of Wang tiles and

x: Z2 → T is a tiling, then for (m,n) ∈ Z2, we define σ(m,n)XT

: XT → XT by

σ(m,n)XT

(x)(i, j) = x(i + m, j + n).

Definition 4.26: Let (T, K, n, e, s, w) be a system of Wang tiles. A tiling x : Z2 → T

is called aperiodic if Per(x) = (0, 0).

There is an important question whether a given set of Wang tiles admits (a)

any aperiodic tilings; (b) a dense set of aperiodic tilings; or even (c) only aperiodic

tilings. Before we study these questions, we identify some necessary conditions for

the existence of periodic tilings. Suppose x : Z2 → T has period (m, 0), if we consider

row 0 of x, we have Tfi= x(i, 0) = x(i + m, 0) = Tfi+m

for all i ∈ Z. Then by

Proposition 4.6, the bi-infinite sequence · · · f−2f−1f0f1f2 · · · ∈ XF satisfies fi = fi+m

for all i ∈ Z. In particular f0f1 · · · fm−1 is a loop in F . Similarly, this argument

occurs at each row j of x for all j ∈ Z.

Lemma 4.27: Let (E, F, p, q) be a textile system such that p, q have path lifting.

Let (T, K, n, e, s, w) be the associated system of Wang tiles. For an integer a > 0,

there exists a periodic tiling x : Z2 → T with period (a, 0) if and only if there are

cycles αj : j ∈ Z of length a in F such that q(αj+1) = p(αj) for all j ∈ Z.

Proof. (⇒) Suppose x : Z2 → T is a tiling of period (a, 0). For each i, j ∈ Z,

let fj(i) ∈ F 1 be the edge such that x(i, j) = Tfj(i). For each j ∈ Z, define αj :=

fj(0)fj(1) . . . fj(a − 1). Fix j ∈ Z. Since the tiling x has period (a, 0), we have

x(a + i, j) = σ(a,0)(x(i, j)) = x(i, j) for all i ∈ Z. In particular, x(a, j) = x(0, j) =

Tfj(a), and hence fj(a) = fj(0). Hence

s(αj) = s(fj(a− 1)) = e(x(a− 1, j))

= w(x(a, j)) = r(fj(a)) = r(fj(0)) = r(αj).


So that αj is a cycle. For i = 0, 1, . . . , a − 1, n(x(i, j)) = n(Tfj(i)) = p(fj(i)), and

s(x(i, j +1)) = s(Tfj+1(i)) = q(fj+1(i)). Since x is a valid tiling, n(x(i, j)) = s(x(i, j +

1)), so q(fj+1(i)) = p(fj(i)) for all i ≤ a− 1. Hence q(αj+1) = p(αj) for all j ∈ Z.

(⇐) Fix cycles αj : j ∈ Z of length a in F such that q(αj+1) = p(αj) for all

j ∈ Z. Define x : Z2 → T as follows: for i, j ∈ Z,

x(i, j) = Tαj(m) where i = ka + m and k ∈ Z, 0 ≤ m < a. (4.5)

Fix i, j ∈ Z, and write i = ka + m as in (4.5). Since αj is a cycle, s(αj(m)) =

r(αj(m + 1 mod a)) for all 0 ≤ m < a. Hence, e(x(i, j)) = e(Tαj(m)) = s(αj(m)) and

w(x(i + 1, j)) = w(Tαj(m+1 mod a)) = r(αj(m + 1 mod a)).

Since s(αj(m)) = r(αj(m + 1 mod a)), we have e(x(i, j)) = w(x(i + 1, j)), which

implies that x satisfies the e−w condition.

Similarly, n(x(i, j)) = n(Tαj(m)) = p(αj(m)) and s(x(i, j + 1)) = s(Tαj+1(m)) =

q(αj+1(m)). By hypothesis, q(αj+1) = p(αj), so n(x(i, j)) = s(x(i, j + 1)), which

implies that x satisfies the n− s condition. Hence x is a valid tiling.

By (4.5), for i, j ∈ Z such that i = ka + m, we have i + a = (k + 1)a + m. Thus

x(i, j) = Tαj(i) = Tαj(i+a) = x(i + a, j). Hence x(i, j) = x(i + a, j) for all i, j ∈ Z, so

the tiling x has period (a, 0). 2

Lemma 4.28: Let (E, F, p, q) be a textile system such that p, q have r-path lifting

and s-path lifting. Let (T, K, n, e, s, w) be the associated system of Wang tiles. Let b

be an integer bigger than 0. There exists a tiling x : Z2 → T with period (0, b) if and

only if there are bi-infinite paths α0, α1, . . . , αb−1 in F such that

p(αj(i)) = q(α(j+1mod b)(i)) for all i ∈ Z and 0 ≤ j ≤ b− 1.

Proof. (⇒) Suppose x : Z2 → T is a tiling of period (0, b). Define Tαj(i) = x(i, j)

whenever j = 0, 1, · · · , b− 1. Fix j ∈ Z. For all i ∈ Z,

s(αj(i)) = e(x(i, j)) = w(x(i + 1, j)) = r(αj(i + 1))

which implies that αj is a bi-infinite path in F . For j ≤ b− 1,

p(αj(i)) = n(x(i, j)) = s(x(i, j + 1)) = q(αj+1(i)).


Hence, p(αj(i)) = q(αj+1(i)). Moreover, since x has period (0, b), we have x(i, b) =

x(i, 0). Then

p(αb−1(i)) = n(x(i, b− 1)) = s(x(i, b)) = s(x(i, 0)) = q(α0(i)).

(⇐) Conversely, suppose that we have bi-infinite paths α0, α1, . . . , αb−1 in F sat-

isfying p(αj(i)) = q(α(j+1mod b)(i)) for all i ∈ Z and 0 ≤ j ≤ b− 1. Define x : Z2 → T

as follows: for i, j ∈ Z,

x(i, j) = Tαm(i) where j = ka + m and k ∈ Z, 0 ≤ m ≤ b− 1. (4.6)

We must check that (i, j) → x(i, j) is a tiling with period (0, b). Since e(x(i, j)) =

s(αm(i)) and w(x(i + 1, j)) = r(αm(i + 1)), and since αm is a bi-infinite path, then

r(αm(i + 1)) = s(αm(i)). Hence e(x(i, j)) = w(x(i + 1, j)), which implies that x

satisfies the e–w condition.

Since n(x(i, j)) = n(Tαm(i)) = p(αm(i)) and s(x(i, j + 1)) = s(Tα(m+1 mod b(i)) =

q(α(m+1mod b(i)), and since p(αm(i)) = q(αm+1mod b(i)), then we have n(x(i, j)) =

s(x(i, j + 1)), which implies that x satisfies the n–s condition. So, x is a tiling.

By (4.6), j = kb + m satisfies x(i, j) = Tαm(i). Since j + b = (k + 1)b + m, we

obtain x(i, j + b) = x(i, j) = Tαm(i). Hence, x(i, j) = x(i, j + b) for all i, j ∈ Z, so x

has period (0, b). 2

Remark 4.29: Let (E, F, p, q) be a textile system, and let (T, K, n, e, s, w) be the

associated system of Wang tiles. For each tiling x : Z2 → T and each j ∈ Z, there is

a unique bi-infinite path αxj in F∞ such that x(i, j) = Tαx

j (i) for all i.


Let (T, K, n, e, s, w) be the associated system of Wang tiles.

(1). Suppose x : Z2 → T is a tiling with period (a, b). Then the element

α0, α1, . . . , αb−1 in XF defined by Tαj(i) = x(i, j) satisfy p(αj) = q(αj+1) for all

0 ≤ j < b− 1 and p(αb−1(i)) = q(α0(i− a)) for all i ∈ Z.

(2). Suppose that α0, α1, . . . , αb−1 ∈ XF satisfy p(αj) = q(αj+1) for all 0 ≤ j <

b− 1 and p(αb−1(i)) = q(α0(i− a)) for all i ∈ Z. For i, j ∈ Z, write j = kb + m where

0 ≤ m ≤ b− 1, and define x(i, j) := Tαm(i−ka). Then x is a tiling with period (a, b).


Proof. (1) Write j = kb + r with k ∈ Z and 0 ≤ r ≤ b− 1. If k ≥ 0, then

x(i− ka, j − kb) = x(i− (k − 1)a, j − (k − 1)b) = . . . = x(i, j). (4.7)

If k < 0, we claim x(i− ka, j − kb) = x(i, j) for all i, j ∈ Z. Fix i, j. Then by (4.7),

since −k ≥ 0,

x(i, j) = x((i− ka)− (−k)a, (j − kb)− (−k)b) = x(i− ka, j − kb).

For 0 ≤ j ≤ b−1 and i ∈ Z, define αj(i) to be the edge of F such that x(i, j) = Tαj(i).

Then by Remark 4.29, αj ∈ XF . Fix 0 ≤ j < b− 1. Then

p(αj(i)) = n(x(i, j)) = s(x(i, j + 1)) = q(αj+1(i)) for all i ∈ Z.

Hence, q(αj(i)) = p(αj+1(i)). Moreover,

x(i, b) = x((i− a) + a, 0 + b) = x(i− a, 0) = Tα0(i−a) for all i.

So for all i ∈ Z,

p(αb−1(i)) = n(x(i, j)) = s(x(i, j + 1) = s(x(i− a, 0) = q(α0(i− a)).

(2) For i, j ∈ Z, write j = kb + m where 0 ≤ m ≤ b − 1, and define x(i, j) :=

Tαm(i−ka). We must check that x is a tiling with period (a, b). Fix i, j ∈ Z. Then

e(x(i, j)) = s(αm(i− ka)) = r(αm(i− ka+1)) = r(αm((i+1)− ka)) = w(x(i+1, j)).

Hence, e(x(i, j)) = w(x(i + 1, j)), which implies that x satisfies the e–w condition.

We have n(x(i, j)) = p(αm(i − ka)). To calculate s(x(i, j + 1)), there are 2 cases to

consider: either 0 < m < b− 1, or m = b− 1.

Case 1: If 0 < m < b− 1, then

s(x(i, j + 1)) = q(αm+1(i− ka)) = p(αm(i− ka)) = n(x(i, j)).

Case 2: If m = b− 1. Then

s(x(i, j + 1)) = s(x(i, kb + (m + 1)))


= s(x(i, (k + 1)b)

= s(x(i− a, kb)

. . .

= s(x(i− (k + 1)a, 0)

= q(α0(i− (k + 1)a)).

Hence,

n(x(i, j)) = p(αb−1(i− ka)) = q(α0(i− (k + 1)a)) = s(x(i, j + 1)).

Then, n(x(i, j)) = s(x(i, j + 1)), which implies that x satisfies the n–s condition. By

definition, x(i, j) = Tαm(i−ka), then

x(i + a, j + b) = Tαm((i+a)−(k+1)a) = Tαm(i−ka) = x(i, j).

Hence x(i + a, j + b) = x(i, j) for all i, j ∈ Z, and so x has period (a, b). 2


Suppose that |F 0| < ∞ and suppose that α is a cycle in F . Then

(1) there exists a cycle α+ ∈ F such that q(α+) = p(αn) and n := |α+||α| belongs to

1, . . . , |F 0| ⊆ N;

(2) there exists a cycle α− ∈ F such that p(α−) = q(αm) and m := |α−||α| belongs

to 1, . . . , |F 0| ⊆ N;

(3) if p, q have unique path lifting, then for all v, w ∈ F 0 such that q(v) = p(r(α))

and p(w) = q(r(α)), there exist unique shortest cycles α+, α− ∈ F satisfying (1) and

(2) such that r(α+) = v and r(α−) = v.

Proof. (1) The cycle α determines a bi-infinite path (α)∞ ∈ XF . Then β =

p(α) ∈ E∗ satisfies r(β) = s(β) = p(r(α)) = p(s(α)) =: u. Thus, the bi-infinite path

(β)∞ ∈ XE satisfies p((α)∞) = (β)∞.

We have v ∈ F 0 : p(v) = u ≤ |F 0| < ∞. Since q is onto, Lemma 3.2 gives a bi-

infinite path γ = . . . γ(−1)γ(0)γ(1) . . . such that q(γ) = β∞, |γ(i)| = |α| and q(γ(i)) =

β for all i ∈ Z. Since F 0 is finite, there exist i < j such that r(γ(i)) = r(γ(j)) but

r(γ(i)) 6= r(γ(i + k)) for all 0 < k ≤ j − i − 1 ≤ |F 0|. Now let n = j − i and define


α+(t) = γ(t + i − 1) for all 1 ≤ t ≤ n. Define α+ := α+(1)α+(2) . . . α+(n) ∈ F n|α|.

We claim that α+ is a cycle. Since

r(α+) = r(α+(1)) = r(γi) = r(γj) = s(γj−1) = s(α+(n)) = s(α+),

α+ is a cycle. So the claim is confirmed, which gives (1).

(2) The same argument with the roles of p and q reversed gives a cycle α− ∈ Fm|α|

satisfying p(α−) = q(αm) and m := |α−||α| belongs to 1, . . . , |F 0| ⊆ N.

(3). If p, q have unique path lifting, fix v ∈ F 0 such that q(v) = p(r(α)).

By Lemma 3.2, since q has unique path lifting, there exists a unique path γ =

. . . γ(−1)γ(0)γ(1) . . . ∈ XF such that q(γ(i)) = β and r(γ(0)) = v. Moreover, from

the proof of part (1), there exist i < j such that γ(i) . . . γ(j − 1) is a cycle and

γ(i) . . . γ(i+k) is not a cycle for k < j− i−1. Now we claim that γ(0) . . . γ(j− i−1)

is also a cycle. To see this, since s(γ(i− 1)) = r(γ(i)) = s(γ(j − 1)), by Lemma 3.2,

unique path lifting forces that γ(2i−j) . . . γ(i−1) and γ(i) . . . γ(j−1) are equal. Then

inductively, we have γ(ki−(k−1)j) . . . γ((k−1)i−(k−2)j−1) = γ(i) . . . γ(j−1) for all

k ∈ Z. Then there exists k such that γ(0) ∈ γ(ki−(k−1)j) . . . γ((k−1)i−(k−2)j−1).

Then γ(0) . . . γ(j−i−1) is also a cycle such that s(γ(0) . . . γ(j−i−1)) = r(γ(0)) = v.

Let α+ = γ(0) . . . γ(j− i−1). The unique path lifting implies that γ(0) . . . γ(j− i−1)

is the unique element of vF k|α| such that q(γ(0) . . . γ(k)) = βk. So γ(0) . . . γ(k) is not

a cycle for k < j− i−1 by construction. So α+ is the unique shortest cycle satisfying

(1). Considering q, a similar argument gives a unique shortest cycle α− satisfying (2).

So all above demonstrate (3). 2

Proposition 4.32: Let (E, F, p, q) be a textile system such that p, q have unique

path lifting. Let (T, K, n, e, s, w) be the associated system of Wang tiles. Suppose that

|F 0| < ∞ and suppose that α is a cycle in F . Suppose that there exist integers b > 1,

a ≥ 0 and bi-infinite paths α0, α1, . . . , αb−1 in F with α0 = (α)∞, p(αj) = q(αj+1) for

all 0 ≤ j < b− 1 and p(αb−1(i)) = q(α0(i− a)) for all i. Then there exist a tiling x,

and integers a′ ≤ |F 0|b−1|α|, b′ ≤ |F 0|b−1|α| · ab such that

i). x has period (a, b);

ii). x has period (a′, 0); and

iii). x has period (0, b′).


Proof. i). Lemma 4.30.(2) applied to the bi-infinite paths α0, α1, . . . , αb−1 implies

that there exists a tiling x with period (a, b).

ii). By Lemma 4.31 (3), we can find a unique cycle α+ ∈ F such that q(α+) =

p(αn1), n1 := |α+||α| belongs to 1, . . . , |F 0| ⊆ N and r(α+) = r(α1(0)). Since q has

unique path lifting, then by Lemma 3.2, α1 = (α+)∞. Similarly, we can also find cycle

α++ such that α2 = (α++)∞. Proceeding this way, we obtain cycles λ1, λ2, . . . , λb−1

and integers n1, . . . , nb−1 ≤ |F 0| such that αi = (λi)∞ for each i, and |λi+1| = ni+1 ·

|λi| = n1n2 . . . ni+1|α| for each i.

Observe that each αi has period n1n2 . . . nb−1|α| for all 0 ≤ i ≤ b− 1. For j ∈ Z,

write j = kb+m with k ∈ Z and 0 ≤ m ≤ b−1, and define x(i, j) := Tαm(i−ka). Since

each αi has period n1n2 . . . nb−1|α|, we have

x(i, j) = x(i + n1n2 . . . nb−1|α|, j) for all i, j ∈ Z.

Hence, x has period (a′, 0) where a′ = n1n2 . . . nb−1|α| ≤ |F 0|b−1|α|.iii). Let M = lcm(n1n2 . . . nb−1|α|, a). We claim that x(i, j + b ·M) = x(i, j) for

all i, j ∈ Z. To see this, fix i, j ∈ Z. Then

x(i, j + b ·M) = x(i− a, j − b + b ·M)

= x(i− 2a, j − 2b + b ·M)

= x(i− 3a, j − 3b + b ·M)

. . .

= x(i−M · a, j)

= x(i− lcm(n1n2 . . . nb−1|α|, a) · a, j).

By definition of x, we have x(i, j) = Tαm(i−ka) for all i, j, and since all αm have

period n1n2 . . . nb−1|α|, we have

x(i− lcm(n1n2 . . . nb−1|α|, a) · a, j) = x(i, j). (4.8)

Hence x(i, j + b · M) = x(i, j) for all i, j. Then the claim is confirmed. Thus x is

(0, b′)-periodic where

b′ = b ·M = b · lcm(n1n2 . . . nb−1|α|, a) ≤ |F 0|b−1|α| · ab. 2


Proposition 4.33: Let (E, F, p, q) be a textile system such that p, q have path lifting.

Let (T, K, n, e, s, w) be the associated system of Wang tiles. Suppose that |E0| =

|F 0| < ∞ and |F 1| < ∞. If there exists a cycle in F , there exist a ≤ |F1|, b ≤ |F 1||F 1|

and a tiling x : Z2 → T such that x has period (a, b).

Proof. Since |E0| = |F 0| < ∞ and since p, q are surjective, p and q are also

injective on F 0. Let α1 be a simple cycle in F , and let u1 = r(α1) = s(α1). Then

|α1| ≤ |F1|. The path β1 := p(α1) ∈ E satisfies p(r(β1)) = p(s(β1)) = p(u1). Hence,

β1 is a cycle in E and is a simple cycle as p is injective. Let v1 := r(β1). Then (α1)∞

is a bi-infinite path in F , and p((α1)∞) = (β1)

∞ ∈ E∞. Since q is injective on F 0 and

has path lifting, there exists a cycle α2 ∈ F , which is necessarily a simple cycle, such

that q(α2) = β1. We then have q((α2)∞) = (β1)

∞. Proceeding in this way, we obtain

simple cycles αi in F and βi in E such that p(αi) = βi = q(αi+1) for all i. Since

|F 1| < ∞, there exist distinct i, j ≤ |F 1||α1| such that αi = αj. Since |α1| ≤ |F 1|, and

|αi| = |αj| for all i, j, we then have i, j ≤ |F 1||F 1|.

Let a := |α1| and let b := j− i. Define x : Z2 → T by x(m,n) = Tα

(n mod a)(m mod b)

. Then

x is a tiling and has a period (a, b) by construction. 2

Let (E, F, p, q) be a textile system. Suppose that (E, F, p, q) has a periodic tiling.

We can not conclude that p, q have r-path lifting and s-path lifting.

Recall Example 2.11 in which p, q have r-path lifting and s-path lifting, and

(E, F, p, q) has a periodic tiling x with Per(x) = (2, 0). Consider the graphs of

Figure 4.7. Add an edge m in F and an edge c in E as illustrated, and set p(m) =

c = q(m). There still exists periodic associated system of Wang tile. But there exists

no edge f in F 1 such that r(f) = w and p(f) = c, so p does not have r-path lifting.

Similarly, q does not have r-path lifting or s-path lifting.

v w

l

h

g k:F

uab

:E


Figure 4.7. The textile system has a periodic tiling and p, q have path lifting.

v w

l

h

g k:F

ua b:E

c

m

z

Figure 4.8. The textile system still has a periodic tiling, but p, q do not have r-path

lifting or s-path lifting.

5. APERIODICITY OF SHIFT SPACES

5.1 Aperiodicity

As discussed in Chapter 4, not all tilings for a textile system (E, F, p, q) are

periodic. In this chapter, we will discuss the aperiodicity of the Wang tile system

(T, K, n, e, s, w) associated to a textile system, and give sufficient conditions under

which the shift space associated to such a tiling system is topologically free. Our

approach is based on Robertson and Steger’s ideas [56] (with r = 2).

Definition 5.1: Let X be a shift space. If every open set U ⊆ X contains an element

x which is aperiodic, then we say that X is topologically free.

Let S be a finite set. Following [56], an S × S 0, 1-matrix M is a function

M : S × S → 0, 1, regarded as a matrix indexed by S. So for a, b ∈ S, M(a, b) is

the (a, b)-entry of M . We denote the set of all S × S 0, 1-matrices by MS(0, 1).For i ≤ j ∈ Z, let [i, j] denote the set i, i + 1, i + 2, . . . , j. When m,n ∈ Z2,

recall the notation m ≤ n if mi ≤ ni for i = 1, 2, and when m ≤ n, let [m,n] :=

[m1, n1]× [m2, n2] ⊆ Z2. For M1,M2 ∈ MS(0, 1) and m,n ∈ Z2 with m ≤ n, let

W[m,n] = w : [m,n] → S; Mi(w(l), w(l + ei)) = 1 whenever l, l + ei ∈ [m,n], i = 1, 2.

Note: Our convention for W[m,n] reverses the direction from that of [56] to accom-

modate our definition of paths in directed graphs.

For m ≥ 0 in Z2, let Wm = W[0,m]. We say that an element w ∈ Wm has shape m.

Define the origin and terminus maps o : Wm → S and t : Wm → S by t(w) = w(0)

and o(w) = w(m) respectively. Let W =⋃

m∈N2

Wm. Define d(w) = n if and only if

w ∈ Wn. For w ∈ Wp and m ≤ n ≤ p, we write w|∗[m,n] for the element of Wn−m

defined by w|∗[m,n](q) = w(q + m) for all q ≤ n − m. As in [50], the * is there to

remind us that w|∗[m,n] is not actually the restriction of w to [m,n], but a “shift” of

5. Aperiodicity of shift spaces 81

this restriction: technically, w|∗[m,n] = σm(w)|[0,n−m]. Let w ∈ Wm and p ∈ Z2. Define

σpw : [p, p + m] → S by (σpw)(p + i) = w(i) for i ∈ [0,m]. Let w ∈ Wm and m ≥ 0.

If 0 ≤ p ≤ m, we say w is p-periodic if σpw|[0,m]∩[p,p+m] = w|[0,m]∩[p,p+m].

Suppose that the matrices Mi satisfy the following conditions from [56]:

(H0) Each Mi is a nonzero 0, 1-matrix;

(H1) Let u ∈ Wm and v ∈ Wn. If o(u) = t(v) then there exists a unique w ∈ Wm+n

such that

w|[0,m] = u, and w|∗[m,m+n] = v;

(H2) Consider the directed graph E which has a vertex for each a ∈ S and a

directed edge from a to b for each i such that Mi(a, b) = 1. Then this graph is

strongly connected in the sense that given a, b ∈ S, there is a path in E from a to b.

(H3) For each p ∈ Zk\0. There exists w ∈ W which is not p-periodic.

In [56], Robertson and Steger show that the following three conditions on the Mi

are equivalent to (H1):

(H1a) MiMj = MjMi for all i, j;

(H1b) For i ≤ j, MiMj is a 0, 1-matrix;

(H1c) For i ≤ j ≤ k, MiMjMk is a 0, 1-matrix.

Definition 5.2: Define a category Λ = Λ(W,d) by Obj(Λ) = S = W0 and Mor(Λ) =

Wn for all n ∈ Nk with r(w) := t(w) and s(w) := o(w) respectively. The composition

of morphisms is defined as follows: If µ, ν ∈ Λ and s(µ) = r(ν), µ ν := µν,

where µν is the unique element (given by (H1)) of Wm+n such that (µν)|[0,m] =

µ, and (µν)|[m,m+n] = ν. The identity morphism ida associated to a ∈ Obj(Λ) = S is

the unique function from [0, 0] = 0 to S such that ida(0) = a.

Theorem 5.3: If M1,M2 satisfy conditions (H0) and (H1), then Λ(W,d) is 2-graph

such that Λ(W,d)n = Wn for all n ∈ N2, r(w) = w(0) and s(w) = n for all w ∈ Wn,

and if w ∈ Wn, w′ ∈ Wm and s(w) = r(w′), then ww′ is the unique element of Wn+m

given by (H1).

Proof. To prove that Λ(W,d) is a 2-graph, we show that composition is associative.

That is: if µ, ν, ω ∈ Mor(Λ) satisfy s(µ) = r(ν) and s(ν) = r(ω), then µ (ν ω) =


(µν)ω. We follow the proof of Lemma 1.2 in [56]: Fix 0, 1-matrices Mi satisfying

(H1a),(H1b) and (H1c). Let 1 ≤ i ≤ r. Let w ∈ Wm and choose a ∈ S such that

Mi(a, t(w)) = 1. There exists a unique word v ∈ Wm+eisuch that v|[0,m] = w and

t(v) = a. The picture below illustrates the words v1 and v2 obtained in this way with

i = 1 and i = 2.

0

m1

a m e

1e

[0, ]mW

0

m

2a m e

2e

[0, ]mW

Figure 5.1. The unique words vi ∈ Wm+ei.

Let w1 = µ (ν ω) and w2 = (µ ν)ω. Then w1 and w2 satisfy w1|[0,d(µ)] = µ =

w2|[0,d(µ)], w1|∗[d(µ),d(µν)] = ν = w2|∗[d(µ),d(µν)] and w1|∗[d(µν),d(µνω)] = ω = w2|∗[d(µν),d(µνω)]

We show that w1 = w2. To see this, we apply (H1) twice. The morphisms α1 :=

w1|[0,d(µν)] and α2 := w2|[0,d(µν)] satisfy α1|[0,d(µ)] = µ = α2|[0,d(µ)] and α1|∗[d(µ),d(µν)] =

ν = α2|∗[d(µ),d(µν)], so (H1) implies that α1 = α2. Now w1|[0,d(µν)] = α1 = w2|[0,d(µν)] and

w1|∗[d(µν),d(µνω)] = ω = w2 |∗[d(µν),d(µνω)], so (H1) gives w1 = w2.

To see that Λ is a countable category, recall that S is a finite set, and each [0,m]

is a finite set, so each Wm ⊆ S[0,m] is finite. Since Nk is countable, Λ :=⋃

m∈Nk

Wm is

countable.

Now we show that Λ is a 2-graph. Define d : Λ → N2 by d(λ) = m if and

only if λ ∈ Wm. We must check the factorization property. Fix w ∈ Wm+n , we

must show that there exist unique u ∈ Wm and v ∈ Wn such that w = uv with

o(u) = t(v), u ∈ d−1(m) and v ∈ d−1(n). Define u = w|[0,m] and v = w|∗[m,m+n]. Then

o(u) = u(m) = w(m) and t(v) = v(0) = w(m). By definition, uv ∈ wm+n satisfies


uv|[0,m] = u and uv|∗[m,m+n] = v. Since w has the same properties, (H1) forces w = uv.

Hence Λ(W,d) is a 2-graph with the desired properties. 2

Here, we want to check the conditions of (H0)–(H3) for some examples.

Example 5.4: Recall the Example 3.24: let E, F be the directed graphs illustrated

below

1u

1f:F

:E

2u

2f

3u

3f

1v

1e

2v

2e

Figure 5.2. Textile system (E, F, p, q).

Recall the definitions of morphisms p, q as follow:

p(u1) = v1, p(u2) = v1, and p(u3) = v2,

q(u1) = v2, q(u2) = v1, and q(u3) = v1.

Since p, q have unique path lifiting, by Theorem 3.8, there exists an associated

2-graph Λ = Λ(E, F, p, q). We draw the skeleton of Λ as follows

1v

2v:G

1u

2u

1e

2e

3u

Figure 5.3. Skeleton of 2-graph Λ = Λ(E, F, p, q).

Define matrices M1 and M2 by:

(M1)i,j = |viΛe1vj| and (M2)i,j = |viΛ

e2vj|.

We call M1 and M2 the transition matrices of G. Thus the transition matrices

M1 and M2 are:


M1 =

(1 00 1

)and M2 =

(1 11 0

).

Then

M1M2 =

(1 00 1

)(1 11 0

)=

(1 11 0

)= M2M1. (5.1)

So M1 and M2 are nonzero 0, 1-matrices and (H0), (H1a) and (H1b) hold. Since

(H1c) is trivial for k = 2, (H1) holds. The skeleton G of the 2-graph Λ(E, F, p, q)

is strongly connected by inspection, and is precisely the graph described in (H2), so

(H2) holds. By (5.1), M1M2 = M2M1 is (2× 2) 0, 1-matrix.

We claim that this example does not satisfy (H3). Fix p ∈ Z. We will show

that for any w ∈ W , w is (p, 0)-periodic. Fix l ∈ N2 such that l ≥ (p, 0) and

fix m ∈ [0, l]. By definition of M1 and w, we have w(m + (1, 0)) = w(m). Hence

w(m + (i, 0)) = w(m) for all 0 ≤ i ≤ p. Hence w|[(p,0),l] = w|[2(p,0),l+(p,0)] for all

l. Then w|[0,l]∩[(p,0),l+(p,0)] = σ(p,0)w|[0,l]∩[(p,0),l+(p,0)]. Therefore, w|[0,(p,0)+l] is (p, 0)-

periodic. Since p-periodicity is invariant with the replacement of p by −p, we deduce

that there exists w ∈ W which is (p, 0)-periodic for all p ∈ Z, which implies that (H3)

fails.

Let Γ = Λ(W,d). To illustrate how to check aperiodicity for tiling systems, we

show that Γ is (0, q)-aperiodic for all q 6= 0. Fix (0, q) ∈ N2. We must show that there

exists w ∈ W which is not (0, q)-periodic. Fix m ∈ N2. If w(m) satisfies w(m) = v1,

then w(m+(0, 1)) could be either of v1, v2; but if w(m) = v2 then w(m+(0, 1)) = v1.

Fix l ∈ N2 such that l ≥ (0, q). Construct w as follows:

Let w(0, q) = v1. We consider two cases:

Case 1: When q = 1.

Then w(0, 1) = v1. Let w(0, 2q) = w(0, 2) = v2. Then w(0, q) = v1 6= v2 =

w(0, 2q). Thus w|[(0,q),l] = w|[(0,1),l] 6= w|[(0,2),l+(0,1)] = w|[2(0,q),l+(0,q)]. Hence we have

w|[0,l]∩[(0,q),l+(0,q)] 6= σ(0,q)w|[0,l]∩[(0,q),l+(0,q)].

Case 2: When q > 1.

Let w((0, q) + (0, k)) = v1 for all 0 ≤ k ≤ q − 1 and let w(0, 2q) = v2. Then

w|[(0,q),l] 6= w|[2(0,q),l+(0,q)] and w|[0,l]∩[(0,q),l+(0,q)] 6= σ(0,q)w|[0,l]∩[(0,q),l+(0,q)].


So for any q ∈ N, there exists w ∈ Wn such that

w|[0,l]∩[(0,q),l+(0,q)] 6= σ(0,q)w|[0,l]∩[(0,q),l+(0,q)].

Then w is not (0, q)-periodic. Again invariance of (0, q)-periodicity with q and −q

shows that (H3) is checked.

Example 5.5: There is a second way to obtain 0, 1-matrices for the data in Ex-

ample 5.4. This was discussed by Deaconu [13]. The corresponding two-dimensional

shift over Z2 has the alphabet S ′ = f1, f2, f3. Define the transition matrices M ′1

and M ′2 by:

(M ′1)i,j =

1, if fi ∈ (p−1 q)(fj) ;0, otherwise .

(M ′2)i,j =

1, if r(fj) = s(fi) .0, otherwise .

Then, the transition matrices M ′1 and M ′

2 are:

M ′1 =

0 0 11 1 01 1 0

and M ′

2 =

1 0 00 1 00 0 1

.

.

Since M ′1M

′2 =

0 0 11 1 01 1 0

1 0 00 1 00 0 1

=

0 0 11 1 01 1 0

= M ′

2M′1, the prod-

uct M ′1M

′2 is (2×2) 0, 1-matrix and the transition matrices M ′

1,M′2 commute. Simi-

larly, conditions (H0) and (H1) hold for M ′1,M

′2. Moreover, (M ′

1M′2)

3 =

1 1 12 2 12 2 1

,

which has all entries nonzero. Thus in the associated graph as in (H2), there is at

least one path of length 6 from v to w for all v.w. Hence (H2) holds.

Before the discussion of the next example, we recall Definition 3.42 of the dual

graph construction [2]. This is the higher-rank analog pΛ of the dual graph construc-

tion for directed graphs.

In [2, Proposition 3.2], the authors prove that (pΛ, dp) is a k-graph. We will use

the idea of dual graphs to study the following example:

Example 5.6: Consider the Example 2.5 in [13], pictured below


v w

l

h

:F ua:E

p

q

Since p, q have unique path lifting, Theorem 3.8 gives an associated 2-graph Λ =

Λ(E, F, p, q), and the complete and associative collection of squares is the set C such

that:

hC vw

a

a

lCv w

a

a

So that wa = av are the two factorization of the square Ch, and va = aw are the

two factorization of the square Cl. The skeleton of 2-graph Λ is given as

uav w

The skeleton of associated dual 2-graph (1, 1)Λ is illustrated as follows

av aw

avw

awa

ava

awv

avv aww

Let S = Λ(1,1) = av, aw. Define (2× 2) matrices M1,M2 by:

(M1)µ,ν =

1, if µα = βν for some α, β ∈ Λ(1,0) ;0, otherwise .


(M2)µ,ν =

1, if µα = βν for some α, β ∈ Λ(0,1) .0, otherwise .

This is the construction of Example 5.6 applied to the dual 2-graph (1, 1)Λ de-

scribed in [2]. The commuting transition 0, 1-matrices over S are:

M1 =

(0 11 0

), and M2 =

(1 11 1

). We can check the conditions (H0)–(H2).

Since

M1M2 =

(0 11 0

)(1 11 1

)=

(1 11 1

)= M2M1,

M1M2 = M2M1 is (2× 2) 0, 1-matrix and M1,M2 commute. Thus, (H0) and (H1)

hold. Moreover, the dual 2-graph (1, 1)Λ is strongly connected by inspection of the

skeleton of the associated dual 2-graph, so (H2) holds.

However, this example does not satisfy (H3). We claim that Γ = Λ(W,d) is

(p, 0)-periodic for p = 2k for all k ∈ Z. Fix k ∈ N and let p = 2k. Fix w ∈ W

and l ∈ N2 with l ≥ (2k, 0). Fix m ∈ [0, l]. If w(m) = av, then by definition of

M1, we have w(m + (1, 0)) = aw and then w(m + (2, 0)) = av = w(m). Following

this process, w(m + (2k, 0)) = av = w(m) and w(m + (2k + 1, 0)) = aw for all

k ∈ N. For the same reason, if w(m) = aw, w(m + (1, 0)) = av and w(m + (2, 0)) =

aw = w(m). Hence w(m + (2k, 0)) = aw = w(m) and w(m + (2k + 1, 0)) = av.

Therefore w|[0,l]∩[(0,p),l+(0,p)] = σ(p,0)w|[0,l]∩[(0,p),l+(0,p)]. Hence w is (p, 0)-periodic. Since

p-periodicity is invariant with the replacement of p by −p, the 2-graph Γ is (p, 0)-

periodic for all p ∈ 2Z. Hence (H3) fails.

It is not difficult, on the other hand, to check that Γ is aperiodic with vertical

shifts. To establish this, fix (0, p) ∈ N2. Define l ∈ N2 by l ≥ (0, p). We must show

that there exists w ∈ W which is not (0, p)-periodic. Fix m ∈ [0, l]. Construct w as

follows:

Let w(0, p) = av. Then let w((0, p) + (0, k)) = av for all 0 ≤ k ≤ p − 1 and let

w(0, 2p) = aw. Then w(0, p) 6= w(0, 2p) and then w|[(0,p),l] 6= w|[2(0,p),l+(0,p)]. Then

w|[0,l]∩[(0,p),l+(0,p)] 6= σ(0,p)w|[0,l]∩[(0,p),l+(0,p)]. Thus for any p ∈ N, there exists w ∈ Wn

such that

w|[0,l]∩[(0,p),l+(0,p)] 6= σ(0,p)w|[0,l]∩[(0,p),l+(0,p)],

that is, w is (0, p)-aperiodic. Hence Γ(W,d) is (0, p)-aperiodic for all p ∈ Z.


If we construct transition matrices via the construction of Example 5.5, the com-

muting transition 0, 1-matrices over alphabet S ′ = h, l are M ′1 =

(1 11 1

),M ′

2 =(

0 11 0

). One can also check that these matrices satisfy (H0)–(H2). In fact, M ′

2,M′1

are same as M1,M2, and Γ′ = Λ′(W,d) constructed from M ′1,M

′2 is isomorphic to Γ

constructed by M1,M2. Then the same argument will give that Γ′ is (p, 0)-periodic

for all p ∈ 2Z and is also (0, p)-aperiodic for all p ∈ Z.

Example 5.7: Consider the Example 2.7 in [13] illustrated below:

:F va:E

p

q

w

c

b u

a

Since p, q do not have unique path lifting, we can not apply Theorem 3.8 to ob-

tain a 2-graph. Indeed construct transition matrices using the construction of Exam-

ple 5.5. The transition 0, 1-matrices over S = a, b, c are M1 =

1 1 11 1 11 1 1

and

M2 =

0 1 10 1 11 0 0

. Hence M1M2 =

1 1 11 1 11 1 1

0 1 10 1 11 0 0

=

1 2 21 2 21 2 2

and M2M1 =

0 1 10 1 11 0 0

1 1 11 1 11 1 1

=

2 2 22 2 21 1 1

. So M1,M2 do not com-

mute. Hence M1,M2 are not the transition matrices of a rank 2-graph.

Example 5.8: Recall Example 2.11 pictured as bellow:

v w

l

h

g k:F

uab

:E


The maps p, q : F → E are defined by:

p1(g) = p1(k) = a, p1(h) = p1(l) = b;

q1(g) = q1(k) = a, q1(h) = q1(l) = b.

Since p, q have unique path lifting, Theorem 3.8 implies that there is an associated

2-graph Λ = Λ(E, F, p, q). The skeleton of the 2-graph is illustrated as below:

ua b wv

The squares are

a

a

vv

b

b

w

a

a

v

b

b

wwv w

Let S = Λ(1,1) = av, aw, bv, bw. Construct the transition matrices M1,M2 as in

the construction of Example 5.6 applied to the dual graph (1, 1)Λ

bv bw

av aw

Figure 5.5. The skeleton of the associated dual graph (1, 1)Λ.

M1 =

1 0 0 10 1 1 01 0 0 10 1 1 0

and M2 =

1 1 0 01 1 0 00 0 1 10 0 1 1

.


We have

M1M2 =

1 0 0 10 1 1 01 0 0 10 1 1 0

1 1 0 01 1 0 00 0 1 10 0 1 1

=

1 1 1 11 1 1 11 1 1 11 1 1 1

= M2M1,

and so M1,M2 commute, and their product is a (4× 4) 0, 1-matrix. Hence, condi-

tions (H0) and (H1) hold. By inspection, the skeleton of the associated dual graph

(1, 1)Λ is irreducible and is precisely the graph described in (H2), so (H2) holds.

If we construct transition matrices M ′1,M

′2 over the alphabet S ′ = g, h, k, l as

in Example 5.6, the resulting shift is also associated to a 2-graph.

Let λi = bw aw bv (av)i and µi = bv av bw (aw)i; both λi and µi have length

3 + i. Write ρλ for the infinite row-vector λ1λ2λ3 . . . and ρµ for µ1µ2µ3 . . . and write

Bi for the (i + 1)×∞ matrix

ρµ...

ρµ

ρλ

.

Now define φ : N × N → av, aw, bv, bw to be the function represented by the

infinite matrix M =

...B3

B2

B1

. The first few entries of M are

......

......

......

......

......

......

......

......

bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bw aw bv av bw aw bv av av bw aw bv av av av . . .bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bw aw bv av bw aw bv av av bw aw bv av av av . . .bv av bw aw bv av bw aw aw bv av bw aw aw aw . . .bw aw bv av bw aw bv av av bw aw bv av av av . . .

.

We show that Γ = Λ(W,d) is aperiodic and satisfies (H3). To establish (H3), fix

p ∈ Z2. We must show that there exists an element α ∈ W that is not p-periodic.

Fix m ∈ N and let p = (m, 0) with m ∈ N. There exists k ∈ N such that

k∑i=1

(3 + i) ≤ m ≤k+1∑i=1

(3 + i). (5.2)


We claim that for any h ∈ N, the element l = (m + (3 + k) + (3 + k + 1), h) has the

property that α := M |[0,l+(m,0)] is not p-periodic. To see this, fix h ∈ N. Recall that α

is p-periodic if and only if α|[0,l]∩[p,p+l] = σp(α)|[0,l]∩[p,p+l]; that is α|[p,l] = σp(α)|[p,l] =

α|[2p,l+p]. That l ≥ (m + (3 + k) + (3 + k + 1), u) implies that α|[(m,h),l] contains at

least two consecutive blocks λk and λk+1, which implies that λkλk+1 is a block in

α|[(m,h),l]. By the construction of M , λi 6= λj when i 6= j. Hence λk and λk+1 can not

consecutively appear in α|[(2m,h),l+(m,h)]. Therefore α|[(m,0),l] 6= α|[2(m,0),l+(m,0)]. Then

α is (m, 0)-aperiodic.

Each column of M has the form

...γ3

γ2

γ1

where each γj is equal to one of the

(j+1)×1 columns

bv...bvbw

,

av...

avaw

,

bw...

bwbv

, or

aw...

awav

. Then, when p = (0, n),

(n ∈ N), there exists k′ ∈ N such that

k′∑j=1

(1 + j) ≤ n ≤k′+1∑j=1

(1 + j). (5.3)

We claim that for any h ∈ N, the element l = (h, n + (3 + k′) + (3 + k′ + 1)), has the

property that the word β := M |[0,l+(n,0)] is not (0, n)-periodic. The proof is similar to

the proof of (m, 0)-aperiodicity above.

Finally, for p = (m,n) ∈ N2, (the configuration is as below)

0

l

p

l p

2 p

Figure 5.6 . The configuration of ρ|[0,l]∩[p,p+l] and σρ(w)|[0,l]∩[p,p+l].

define l = (m+(3+ k)+ (3+ k +1), n+(1+ k′)+ (1+ k′+1)) where k and k′ satisfy

(5.2) and (5.3) respectively. As above, ρ|[p,l] 6= ρ|[2p,l+p]. Hence, ρ is p-aperiodic. Since


p-periodicity is invariant with substitution of p and −p, if ρ is (m,n)-aperiodic, ρ is

also (−m,−n)-aperiodic. A similar argument applies if p = (−m,n) or p = (m,−n).

Hence condition (H3) holds.

These examples just discussed motivate the following result.

Proposition 5.9: Suppose that(E, F, p, q) is a textile system and that p, q have

unique path lifting. Let Λ be the associated 2-graph, and let (1, 1)Λ be the dual

graph as in [2]. For (1, 1)Λ, construct transition matrices M1,M2 as in Example 5.6.

Suppose that each row of M1 has at least two nonzero entries. Then (1, 1)Λ is (p, 0)-

aperiodic for all p ∈ Z. If each row of M2 has at least two nonzero entries, (1, 1)Λ is

(0, p)-aperiodic for all p ∈ Z.

Proof. If each row of M1 has at least two nonzero entries, for each 1 ≤ i ≤ n,n∑

j=1

aij ≥ 2. Then for each vertex u ∈ (1, 1)Λ0, there exist two different blue edges

eu, fu ∈ (1, 1)Λ(1,0) such that r(eu) = r(fu) = u but s(eu) 6= s(fu).

We claim that: there exists a vertex x ∈ (1, 1)Λ0 and distinct simple cycles α, β

such that r(α) = r(β) = x and |α|, |β| ≥ 1. Too see this, fix u ∈ (1, 1)Λ0. If there

exist two simple blue cycles C1 and C2 such that u ∈ C1 ∩ C2, we are done. Suppose

not. Let v = s(eu) 6= s(fu). Let β be the maximal simple blue path in (1, 1)Λ(1,0)

containing eu. Then r(β) = r(β1) = r(eu) = u, and s(β1) = v. Let s(β) = x. Since β

is the maximal simple blue path containing eu, for any blue edge g whose range is x,

s(g) must already be in β. In particular, s(eg) and s(fg) lie on β, say s(eg) = r(βi),

s(fg) = r(βj) and r(eg) = r(fg) = x with i 6= j. Now egβi . . . β|β| and fgβj . . . β|β| are

distinct simple blue cycles with range y.

In order to prove that the 2-graph is horizontally aperiodic, we must show that

for any p ∈ N, there exist w ∈ W such that w is not (p, 0)-periodic. We proceed as in

Example 5.8. Regard M as a function from N×N to (1, 1)Λ0. From the claim above,

there exist distinct simple blue cycles α, β and u ∈ (1, 1)Λ0 such that r(α) = r(β) = u.

Then construct M as follows: let M0,0 = u. Define the bottom row M0,j of M by

M0,j = a1a2a3 . . . where ai = αβi ∈ Λ(|α|+i|β|,0).


Now fix m ∈ N and let p = (m, 0). Then there exists k ∈ N such that

k∑i=1

(|α|+ i|β|) ≤ m ≤k+1∑i=1

(|α|+ (i + 1)|β|). (5.4)

We now argue as in Example 5.8. By choice of l, w|[p,l] contains at least two consec-

utive blocks ak and ak+1. Since α, β are different simple cycles, then ak 6= ak+1. The

following proof is similar to the proof in Example 5.8. Since akak+1 ∈ w|[p,l], and since

akak+1 /∈ w|[2p,l+p], then w|[(p,l] 6= w|[2p,l+p], so w is not (m, 0)-periodic. Thus (1, 1)Λ

is (p, 0)-aperiodic for all p ∈ Z.

A symmetric argument, using matrix M2 instead of M1, show that (1, 1)Λ is also

(0, n)-aperiodic for all n ∈ Z. 2

In [2], a checkable aperiodicity condition of k-graphs is given as follows:

(H3∗): Fix j, 1 ≤ j ≤ r. Let m ∈ Zr+ with mj = 0. Let w ∈ Wm. Then there

exists u, u′ ∈ Wm+ejsuch that u|[0,m] = u′|[0,m] = w but u(ej) 6= u′(ej).

Recall the definition of aperiodicity of textile systems: Let (E, F, p, q) be a textile

system, and let (T, K, n, s, e, w) be a system of Wang tiles. A tiling x : Z2 → T is

called aperiodic if and only if Per(x) = (0, 0). Let S be a subset of a topological space

X. A shift-space X is topologically free if its aperiodic elements form a dense subset

of X.

In the next theorem, we provide an aperiodicity condition for textile systems

inspired by (H3∗).

Theorem 5.10: Let (E, F, p, q) be a textile system with path lifting. Consider the

the associated system (T, K, n, s, e, w) of Wang tiles. The shift space XT associated

to the tiling system is topologically free if both of the following conditions hold:

(i): For all λ ∈ E∗, there exist paths u, u′ ∈ F ∗ such that q(u) = q(u′) = λ and

r(u) 6= r(u′);

(ii): For every sequence v1, . . . , vn ∈ F 0 such that p(vi) = q(vi+1) for all 0 < i < n,

there exist sequences α1, . . . , αn and β1, . . . , βn in F 1 such that p(αi) = q(αi+1) and

p(βi) = q(βi+1) for all 0 < i < n − 1, and r(αi) = r(βi) = vi for all 0 < i ≤ n but

q(α1) 6= q(β1).


Proof. Write W |[g,h] when g ≤ h ∈ Z2 for the set of functions w : [g, h] → T

satisfying the compatibility conditions in Definition 2.12. Let W =⋃

g≤h

W |[g,h]. We

claim that: Claim 1: for any w ∈ W[g,h] and any p ∈ Z2\(0, 0), there exist l ∈ Z2

and w′ ∈ W[g,h]+l such that w′|[g,h] = w and w′ is not p-periodic. Fix w ∈ W[g,h].

Observe that a tiling is p-periodic if and only if it is (−p)-periodic, so we only need

to check cases when sign of coordinate of p are same or different. We consider this in

two cases:

Case 1: When p /∈ [0,m].

Subcase 1.1: p1, p2 ≥ 0.

Choose w∗ ∈ W[0,m]+p such that w∗|[0,m] = w. If w∗|[p,p+m] 6= w, then w′ = w∗

suffices. So, suppose that w∗|[0,m] = w∗|[p,p+m] = w. Then w(i, j) = w∗((i, j) + p)

for all 0 ≤ (i, j) ≤ m. Let λi = w∗|[(i−1,0),(i,0)] for all i = 1, 2, . . . , m1. Then λ =

λ1λ2 . . . λm1 ∈ E∗. Thus,

λ = w∗|[0,(m1,0)] = w|[0,(m1,0)] = w∗|[p,p+(m1,0)].

By hypothesis (i), there exist u, u′ ∈ F ∗ such that

s(Tui) = q(ui) = λi = q(u′i) for all i = 1, 2, . . . , m1, but

w(Tu1) = r(u) 6= r(u′) = w(Tu′1).

Then there are w′1, w

′2 ∈ W[0,m]+p such that

w′1|[0,m] = w and w′

1|[p,p+(m1,0)] = u,

w′2|[0,m] = w and w′

2|[p,p+(m1,0)] = u′.

However, r(u) = w(Tw′1(p)) and r(u′) = w(Tw′2(p)) are different. Hence one of

w(Tw′1(p)) and w(Tw′2(p)) must be different to w(Tw(0)). Without lose of generality,

suppose it is w′1 such that w(Tw′1(p)) 6= w(Tw(0)). Then w(Tw′1(p)) 6= w(Tw′1(0)). Thus,

σp(w′|[0,m]) 6= w′|[0,m], which implies that w′1 is not p-periodic.


0

p m

( ')u u

m

p

Figure 5.7. The configuration of the proof of subcase 1.1.

Subcase 1.2: p1 ≥ 0 and p2 ≤ 0.

Choose w∗ ∈ W |[0,m]+p such that w∗|[0,m] = w. If w∗|[p,p+m] 6= w, then w′ = w∗

suffices. Otherwise, suppose that w∗|[0,m] = w∗|[p,p+m] = w. Let v1, . . . , vm2 be a

sequence in F 0 such that

vi = w∗|[(0,i−1),(0,i)] = w|[(0,i−1),(0,i)] = w∗|[p+(0,i−1),p+(0,i)].

These vi satisfy that p(vi) = q(vi+1) for all 0 < i < m2. By hypothesis (ii), there

exist sequences α1, . . . , αm2 and β1, . . . , βm2 in F 1 such that p(αi) = q(αi+1) and

p(βi) = q(βi+1) for all 0 < i < m2 and

w(Tαi) = r(αi) = r(βi) = w(Tβi

) for all i = 1, 2, . . . , m2, but

s(Tα1) = q(α1) 6= q(β1) = s(Tβ1).

Then there are w′1, w

′2 ∈ W[0,m]+p such that for all 0 < i < m2

w′1|[0,m] = w and Tαi

= w′1|[p+(0,i−1),p+(1,i)],

w′2|[0,m] = w and Tβi

= w′2|[p+(0,i−1),p+(1,i)].

Since w′1|[p,p+(1,0)] = q(α1) 6= q(β1) = w′

2|[p,p+(1,0)], one of w′1|[p,p+(1,0)] and w′

2|[p,p+(1,0)]

is different to w|[0,(1,0)]. Without lose of generality. w′1|[p,p+(1,0)] 6= w|[0,(1,0)]. Then

s(Tw′1(p)) 6= s(Tw′1(0)). Hence σp(w′1|[0,m]) 6= w′

1|[0,m], which implies that w′1 is not

p-periodic.


1k

2k

mk

0

m

p

p m

1 1( )T T

1k

2k

mk

2 2( )T T

( )m m

T T


Case 2: When p ∈ [0,m].

Subcase 2.1: If w(m− p) 6= w(m).

Then we can choose any extension of w′ of w. Since w(m − p) 6= w(m), then w′

is not p-periodic.

Subcase 2.2: If w(m− p) = w(m).

Construct w∗|[m,m+p] as in case 1. Then r(w|[m−p,m]) = s(w|[m−p,m]) = r(w∗|[m,m+p])

but w∗|[m,m+p] 6= w|[m−p,m]. Then w′ is the unique word in W |[0,m+p] such that

w′|[0,m] = w and w′|[m,m+p] = w∗|[m,m+p] 6= w|[m−p,m]. Hence w′ is not p-periodic.

0

m

m p

m p



This proves Claim 1.

Now fix w ∈ W∗. List Z2\0 = p1, p2, . . .. Claim 2: for each i ≥ 1, there exists

wi ∈ W such that wi|dom(wi−1) = wi−1 and wi is not pj-periodic for any j < i. We

prove this by induction. Let w0 = w. When i = 1, fix p1 ∈ Z2\0. By Claim 1,

there exists a w1 such that w1|[0,m] = w0 = w, but w1 is not p1-periodic. As an

inductive hypothesis, suppose that there exists wn such that wn|dom(wn−1) = wn−1

but wn is not p1, . . . pn-periodic. Applying Claim 1, there exists wn+1 such that

wn+1|dom(wn) = wn and wn+1 is not pn+1-periodic. Since wn+1|dom(wn) = wn, and since

wn is not p1, . . . pn-periodic, then wn+1 is also not p1, . . . pn-periodic. This proves

Claim 2.

Define x+ : N2 → T as follows: for (m,n) ∈ N2, choose i such that (m,n) ∈dom(wi), and define x+(m,n) := wi(m,n). Then w satisfies condition (1) and (2) of

Definition 2.12. Since p, q have path lifting, there exists x : Z2 → T satisfying s–n

and e–w conditions, and x|N2 = x+. Since x|[0,dom(wi)] = wi for all i, and since each

wi is aperiodic, x is aperiodic. That is, the cylinder set Z(w) contains an aperiodic

element x. Since the Z(w) forms a basis for the topology on XT , this proves that the

aperiodic points are dense in XT . 2

Now, recall Example 5.8. We want to check that this system satisfies the hypoth-

esis of Theorem 5.10.

Firstly, we check condition (i). Fix λ ∈ E∗. We consider two cases: λ = aµ or

λ = bν with µ, ν ∈ E∗. When λ = aµ, since q(g) = q(k) = a, and v = r(g) 6= r(k) =

w, then by r-path lifting, there exist path u, u′ ∈ F ∗ such that q(u1) = q(u′1) = a

but r(u) = v 6= w = r(u′); so u 6= u′. Similarly, when λ = bν, since q(h) = q(l) = b,

and w = r(h) 6= r(l) = v, by r-path lifting, we can also find u, u′ ∈ F ∗ satisfying (i).

Hence, (i) is confirmed.

Secondly, we check condition (ii). Fix a sequence v1, . . . , vm ∈ F 0 such that

p(vi) = q(vi+1) for all 0 < i < m. We discuss four cases as follows: (1) v1 = v2 = v;

(2) v1 = v, v2 = w; (3) v1 = v2 = w and (4) v1 = w, v2 = v.

(1): Suppose that v1 = v2 = v. Let α1 = α2 = l and β1 = β2 = g. Then

p(α1) = q(α2) and p(β1) = q(β2), but q(α1) = b 6= a = q(β1). Since p, q have r-path

lifting, there exist α3, . . . , αm, β3, . . . , βm ∈ F 1 such that r(αi) = r(βi) = ki for each i,


and p(αi) = q(αi+1), p(βi) = q(βi+1) for each i. Thus we have α1 . . . αm and β1 . . . βm

satisfying condition (ii).

(2): Suppose v1 = v and v2 = w. Define α1 = l, α2 = h, β1 = g and β2 = k. By

an argument like that in case (1), we extend them to αm and βm satisfying condition

(ii).

In case (3), let α1 = α2 = k and β1 = β2 = h, and argue as in case (1). Finally,

in case (4), let α1 = h, α2 = k,β1 = l and β2 = h, and argue as in case (2). Hence

Example 5.8 satisfies the hypothesis of Theorem 5.10.

Definition 5.11: Given a 2-dimensional shift space X, we write X∗ for the collection

x|[0,m] : x ∈ X,m > 0 of allowed blocks from X.

Lemma 5.12: Let (E, F, p, q) be a textile system with path lifting. Let (T, k, n, s, e, w)

be the associated system of Wang tiles. Suppose that for any u, v ∈ E0, there exists

λ ∈ E∗ and a sequence v1, v2, . . . , vn ∈ F 0 such that r(λ) = u and s(λ) = q(v1),

p(vi) = q(vi+1) for all 1 ≤ i ≤ n − 1 and p(vn) = v. Then for all w1, w2 ∈ W ∗, there

exists a tiling x : Z2 → T such that x|[0,d(w1)] = w1 and x|[d(w)−d(w2),d(w)] = w2.

Proof. Fix w1, w2 ∈ X∗ such that s(w1) = u ∈ E0 and r(w2) = v ∈ E0. We

want to show that there exists w ∈ X∗ such that r(w(d(w1), d(w) − d(w2)) = u and

s(w(d(w1), d(w)− d(w2)) = v.

By hypothesis, for any u, v ∈ E0, there exists λ ∈ E∗ and a sequence v1, . . . , vn ∈F 0 such that r(λ) = u and s(λ) = q(v1), p(vi) = q(vi+1) for all 1 ≤ i ≤ n − 1 and

p(vn) = v. Using s-path lifting and by Lemma 3.2, there exist α1 ∈ F ∗ such that

s(α1) = v1 and s(Tα1(j)) = n(Tλ(j)) for all 1 ≤ j ≤ |λ|. Then, inductively there exist

α2, . . . , αn ∈ F ∗ such that s(αi) = vi for all 2 ≤ i ≤ n, and p(αi) = q(αi+1) for

all 1 ≤ i ≤ n − 1. Hence there exists w′ ∈ X∗ such that r(w′) = s(w1) = u and

s(w′) = r(w2). By path lifting, there exists w ∈ X∗ such that w ∈ Wd(w1)+d(w′)+d(w2)

such that w|[0,d(w1)] = w1, w|[d(w1),d(w)−d(w2)] = w′ and w|[d(w)−d(w2),d(w)] = w2.

For (m,n) ≤ d(w), define x+(m,n) = w(m,n). Since p, q have path lifting,

by Corollary 4.8, there exists x : Z2 → T satisfying s–n and e–w conditions, and

x|N2 = x+. Then x is a tiling satisfying x|[0,d(w)] = w. Hence x|[0,d(w1)] = w1 and

x|[d(w)−d(w2),d(w)] = w2. 2


Definition 5.13: Let X be a topological space, and f : Z2 → Homeo(X) an action

of Z2 by homeomorphisms of X. Then f is said to be topologically transitive if for

any non-empty open sets A,B ⊂ X, there exists n ∈ Z2 such that fn(A) ∩B 6= ∅.

Proposition 5.14: Let (E, F, p, q) be a textile system. Let (T, k, n, s, e, w) be the as-

sociated system of Wang tiles. If the textile system (E, F, p, q) satisfies the conditions

of Lemma 5.12, then σ is topologically transitive.

Proof. Let XT be the shift space associated to the tiling system (T, k, n, s, e, w).

Let A,B ⊂ XT be non-empty open sets. Then in A, for any point a ∈ A, there exist

r1 > 0 and an open ball B(a, r1) ⊆ A; Similarly, in B, for any point b ∈ B, there

exist r2 > 0 and an open ball B(b, r2) ⊆ B. Let r = minr1, r2. Then, B(a, r) ⊆ A

and B(b, r) ⊆ B. Let w = a(−d− log2 re, d− log2 re), w′ = b(−d− log2 re, d− log2 re).Using Lemma 5.12, we can construct z ∈ T with the conditions (i) and (ii) such that

z|[0,d(w)] = w and z|[d(z)−d(w′),d(z)] = w′. Then, z ∈ A and σd(z)−d(w′)(z) ∈ B. Thus

A ∩ σd(z)−d(w′)(B) 6= ∅. Hence T is topologically transitive. 2

We give an easily checkable example to illustrate that the conditions in Lemma 5.12

are sufficient conditions under which T is topologically transitive.

Example 5.15: Let Λ be a 2-graph as follows.

u v

wy

Since Example 5.15 is strongly connected, it satisfies hypothesis in Lemma 5.12.

We check that Example 5.15 is topologically transitive. Fix open A,B ∈ Λ∆. Choose

Z(λ) ⊆ A and Z(µ) ⊆ B. There exists α ∈ Λ such that r(α) = s(λ) and s(α) = r(µ).

Fix x ∈ Λ∆ such that x(0, d(λαµ)) = λαµ. Then x ∈ Z(µ) ⊆ A. Let n := d(λα).

Then

σn(x)(0, d(µ)) = x(0 + n, d(µ) + n) = x(d(λα), d(λαµ)) = µ.


So σn(x) ∈ Z(µ) ⊆ B. Hence σn(A) ∩B 6= ∅.Observe that there is no blue path λ ∈ E∗ such that s(λ) = v and r(λ) = w, and

likewise there is no red path in Λ from v to w. So for the shift space of Λ(E, F, p, q)

to be topologically transitive, it is not necessary that either the red graph or the blue

graph of this system is strongly connected.

Remark 5.16: If (E, F, p, q) has unique path lifting and E, F are finite, then the

associated 2-graph Λ is cofinal if and only if it is strongly connected. This means, in

term of (E, F, p, q) that for all v, w ∈ E0, there exist a path α ∈ E∗ and a sequence

u1, . . . , un ∈ F 0 such that r(α) = v, q(ui) = p(ui+1) for all i ≤ n− 1, and p(un) = w.

Definition 5.17: [52] A shift space X is irreducible if for any allowed blocks u,v,

there is a point x ∈ X and disjoint sets of coordinates S and T such that xS = u and

xT = v.

Lemma 5.18: If textile system (E, F, p, q) satisfies the conditions in Lemma 5.12,

then corresponding shift space X(E, F, p, q) is irreducible.

Proof. From definition 5.17 and Lemma 5.12, then X(E, F, p, q) is irreducible. 2

Remark 5.19: The definition of irreducible seems over complicated at first sight.

A first guess at what irreducible should means might be that for all allowed blocks

u, v, there exists x ∈ X containing both u and v. We show below that this is

equivalent to topological transitivity: for all open U, V , there exists n ∈ Z2 such that

σn(U) ∩ V 6= ∅. However, as pointed at in [36], there is an unsatisfactory notion

of irreducibility because it is satisfied by the SFT coming from the non-irreducible

graph below:

11

01

00

Corollary 5.20: Let (X, σ) be a 2-dimensional shift space. If X is irreducible, σ is

topologically transitive.


Proof. Let A,B ⊂ X be non-empty open sets. Then in A, for any point a ∈ A,

there exists r1 > 0 such that the open ball Br1(a) ⊆ A. In B, for any point b ∈ B,

there exists r2 > 0 such that the open ball Br2(b) ⊆ B. Choose r = min r1, r2. Let

N = d− log2 re. Let u = a|[(−N2

,−N2

),(N2

, N2

)], and let w = b|[(−N2

,−N2

),(N2

, N2

)]. Since X is

irreducible, then for u,w, there exists x ∈ X and disjoint sets of coordinates S, T such

that x|S = u and x|T = w. Assume that S = (p + (−N2,−N

2), p + (N

2, N

2)) and T =

(q +(−N2,−N

2), q +(N

2, N

2)) with p, q ∈ Z2. Then σ−p(x)|[(−N

2,−N

2),(N

2, N2

)] = u. Let x =

σ−p(x). Hence x|[(−N2

,−N2

),(N2

, N2

)] = u, then x ∈ A. And since σq−px|[(−N2

,−N2

),(N2

, N2

)] =

w, we have x ∈ σp−q(B). So x ∈ A ∩ σp−q(B). Thus, σ is topologically transitive. 2

However, the converse of Corollary 5.20 fails. We will use following examples to

illustrate it.

Example 5.21: Let F be a directed graph as Remark 5.19:

e

f

g

:F u v

We want to show that the associated one dimensional shift space (X, σ) is topo-

logically transitive, but not irreducible. For each n ∈ Z, µ ∈ F ∗, set

Z(µ, n) = x : xn = µ1, . . . , xn+|µ| = µ|µ|.

Let A = Z(µ, n), B = Z(ν, m) be open sets in X such that n,m ∈ Z and µ, ν ∈ F ∗.

We need to find N ∈ Z such that σN(A) ∩B 6= ∅. Discuss this in 2 cases.

Case 1: s(ν) = u. Then ν = gk for some k ∈ N.

Subcase 1.1: r(µ) = u. Let x = g∞νµe∞ where ν appears at the coordinates

(m, d(ν) + m), then x ∈ Z(ν, m). So x ∈ B. We claim that there exists N ∈ Z such

that σN(x) ∈ A, so that x ∈ σ−N(A). Then N := n−m− d(ν)− 1 satisfies. Hence,

σ−N(A) ∩B 6= ∅Subcase 1.2: r(µ) = v. Then µ = el, l ∈ N. Let x = g∞fνµe∞. ν appears at the

coordinates (m, d(ν) + m). Then N := n−m− d(ν)− 2 satisfies σN(x) ∈ A.

Case 2: s(ν) = v.


Subcase 2.1: r(µ) = v. Then µ = el. Let x = . . . νµe∞. Use the argument of

Case 1 to see that N := n−m− d(ν)− 1 satisfies σN(x) ∈ A.

Subcase 2.2: r(µ) = u, then ν = gk2fek1 or ek1 and µ = gk′2fek′1 or gk′2 with

k1, k2, k′1, k

′2 ∈ N ∪ 0.

Subcase 2.2.1: ν = ek1 , k1 ∈ N.

If µ = gk′2 , k′2 ∈ N, let x = g∞µfνe∞. Then N := n − m + 1 + d(µ) satisfies

σN(x) ∈ A. If µ = gk′2fek1 , and k′2, k1 ∈ N ∪ 0, then N := n −m + d(µ) satisfies

σN(x) ∈ A.

Subcase 2.2.2: ν = gk2fek1 .

If µ = gk′2 , let x as x = g∞µνe∞. Then N = n −m + d(µ). If µ = gk′2fek1 , let

x = g∞gmaxk2,k′2femaxk1,k′1e∞. Since ν appears at the coordinates (m,m + d(ν)),

µ appears at the coordinates (m + (k2 − k′2),m + (k2 − k′2) + d(µ)). Then N :=

n−m + k′2 − k2 satisfies σN(x) ∈ A.

Hence, (X, σ) is topologically transitive. However, if U = gf and V = fe, then

U, V are allowed blocks in X, but it is straightforward that are no infinite path x ∈ X

and disjoint sets of coordinates S and T such that xS = U and xT = V . Thus, X is

not irreducible. 2

For 2-dimensional shift spaces, we use following example to show that irreducibil-

ity is still stronger than being topologically transitive.

Example 5.22: Let E, F be the infinite directed graphs as follows:

0e

0f

0g

:F

0'e

0'f

0'g

:E

1e

1f

1g

1e

1f

1g

......

1'e

1'f

1'g

1'e

1'f

1'g ...

...


Define morphisms p, q : F → E by: p(gi) = g′i, p(fi) = f ′i and p(ei) = e′i;

q(gi) = g′i−1, q(fi) = f ′i−1 and q(ei) = e′i−1. Then p, q are surjective graph morphisms,

and the map F 1 → (p(F 1), q(F 1), r(F 1), s(F 1)) is injective in F 1. So (E, F, p, q) is a

textile system. Moreover, p, q have path lifting in (E, F, p, q).

Let A = Z(u, n) and B = Z(w, m) be non-empty open sets in X such that

n,m ∈ Z2 and u,w are allowed blocks. Then in A, for any point a ∈ A, there

exists r1 > 0 such that the open ball Br1(a) ⊆ A. In B, for any point b ∈ B, there

exists r2 > 0 such that the open ball Br2(b) ⊆ B. Choose r = min r1, r2. Let

N = d− log2 re. Let u = a|[(−N2

,−N2

),(N2

, N2

)], and let w = b|[(−N2

,−N2

),(N2

, N2

)]. We discuss

in 3 cases.

Case 1: b|[(−N2

,−N2

)+(N−1,0),(−N2

,−N2

)+(N,1)] = Tgkfor some k ∈ Z.

Subcase 1.1: a|[(−N2

,−N2

),(−N2

,−N2

)+(1,1)] = Tfk. Since p, q have path lifting, there

exists x such that

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)]

and w appears at coordinates (m,m + d(w)). So x ∈ Z(w, m). We need to find

M ∈ Z2 such that σM(x) ∈ A. Let M := n−m− d(w)1 − (1, 0). Then σM(x) ∈ A.


,−N2

),(−N2

,−N2

)+(1,1)] = Tfjwith j 6= k. Using argument of

Subcase 1.1, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1+(0,j−k)+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].

Let M := n−m− d(w)1 − (1, j − k). Then σM(x) ∈ A.


,−N2

),(−N2

,−N2

)+(1,1)] = Tej. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1+(1,0),m+d(w)1+(1,0)+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].

Let M := n−m− d(w)1 − (2, 0). If j 6= k, let M := n−m− d(w)1 − (2, j − k).

Case 2: b|[(−N2

,−N2

)+(N−1,0),(−N2

,−N2

)+(N,1)] = Tfk.


,−N2

),(−N2

,−N2

)+(1,1)] = Tej. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].




,−N2

),(−N2

,−N2

)+(1,1)] = Tfj. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1−(1,0)+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].

Let M := n−m− d(w)1. If j 6= k, let M := n−m− d(w)1 + (0, j − k).


,−N2

),(−N2

,−N2

)+(1,1)] = Tgj. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1−(2,0)+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].

Let M := n−m− d(w)1 + (1, 0). If j 6= k, let M := n−m− d(w)1 + (1, j − k).

Case 3: b|[(−N2

,−N2

)+(N−1,0),(−N2

,−N2

)+(N,1)] = Tek.


,−N2

),(−N2

,−N2

)+(1,1)] = Tej. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].



,−N2

),(−N2

,−N2

)+(1,1)] = Tfj. If j = k, let

x|[m,m+d(w)] = b|[(−N2

,−N2

),(N2

, N2

)] and x|[m+d(w)1,m+d(w)1−(2,0)+d(u)] = a|[(−N2

,−N2

),(N2

, N2

)].

Let M := n−m− d(w)1 + (1, 0). If j 6= k, let M := n−m− d(w)1 + (1, j − k).


,−N2

),(−N2

,−N2

)+(1,1)] = Tgj. Then a|[(−N

2,−N

2),(−N

2,−N

2)+(N,1)] =

(Tgk)k2Tfk

(Tek)k1 and b|[(−N

2,−N

2),(−N

2,−N

2)+(N,1)] = (Tgj

)k′2Tfj(Tej

)k′1 such that k2 + 1 +

k1 = k′2 + 1 + k′1 = N . With similar argument of Subcase 2.2 of Example 5.21, we

can find suitable M satisfying σM(A) ∩B 6= ∅.From above, (X, σ) is topologically transitive. However, when we choose allowed

blocks u,w ∈ X such that u = Tg0Tf0 and w = Tf0Te0 , there do not exist x and

disjoint U and S such that x|U = u and x|S = w. Thus, this two-dimensional shift

space X is not irreducible. 2


5.2 Beatty Sequences

We end this chapter, by giving a textile system which has no periodic tilings based

on Kari’s aperiodic set of Wang tiles. We will use the idea of Beatty Sequences to

build this system.

Beatty sequences [4] were introduced by Beatty, who wrote about them in 1926.

Mathematically, a Beatty sequence is the sequence of integers found by taking the

floor of the positive multiples of a positive irrational number.

Given an arbitrary real number r, brc is the integer part of r: that is, brc :=

maxn ∈ Z : n ≤ r. The number r − brc ∈ [0, 1) is called the fractional part of r.

In this section, we will try to use the idea of Beatty Sequences of numbers to tile the

whole plane.

Fix a real number r. We construct a bi-infinite Beatty sequence A(r) consisting

of the integral parts of the multiples of r; that is, for i ∈ Z,

A(r)i = bi · rc.

The sequence B(r)i of the first difference of the Beatty sequence A(r) is given by

B(r)i = A(r)i − A(r)i−1 = bi · rc − b(i− 1) · rc.

In [20], the bi-infinite sequence B(r)i is called the balanced representation of r. If

k ≤ r ≤ k + 1, then B(r) is a sequence of k′s and (k + 1)′s. For instance,

B(1) = . . . 11111 . . . , B(1

3) = . . . 001001 . . . , and B(

3

4) = . . . 01110111 . . . .

The balanced representation of a rational number is a periodic bi-infinite string.

In [20], Kari introduces sequential machines to define mappings on bi-infinite strings.

Sequential machines were used to implement multiplication of numbers in balanced

representations, and later to show that they can be used to construct tile sets.

A sequential machine [20] is defined as a quadruple M = (K, Σ, ∆, γ) such that

K is a finite set of states, Σ is the input alphabet, ∆ is the output alphabet, and

γ ⊆ K×Σ×∆×K is called the transition set. We can represent a sequential machine

M as a labeled directed graph with vertex set K and an edge from vertex q to vertex

p labeled a, b for each transition (q, a, b, p) in γ.


We use the machine M to compute a relation ρ(M) between bi-infinite sequences

of letters. A bi-infinite sequence α over a tile set T is a function α : Z → T . Given

bi-infinite paths α and β over input and output alphabets, we say α and β are in the

relation ρ(M) if and only if there is a bi-infinite sequence s of M such that for each

i ∈ Z, there is a transition from si−1 to si labeled by αi and βi.

In [11, 20], given a positive rational number q = n/m an r ∈ R, Culik and Kari

constructed a Mealy machine Mq by multiplying balanced representations B(α) of real

numbers by q. The set K of states of Mq is the set of possible values of qbrc − bqrc.Since

qbrc − 1 ≤ qr − 1 < bqrc ≤ qr < q(brc+ 1), we have

−q < qbrc − bqrc < 1.

Since qbrc−bqrc is always a multiple of 1m

, each qbrc−bqrc is among the n+m−1

elements of

S = −n− 1

m,−n− 2

m, . . . ,

m− 1

m.

S is the state set of Mq.

Now we construct the transitions of Mq as follows: there exist state s+qa−b and

a transition from s to it with input symbol a and output symbol b. Through reading

input . . . B(α)i−2B(α)i−1 and producing output . . . B(qα)i−2B(qα)i−1, the machine is

in the state

si−1 = qA(α)i−1 − A(qα)i−1 ∈ S.

The next input B(α)i and the resulting outputs B(qα)i satisfy:

si−1 + qB(α)i −B(qα)i = qA(α)i−1 + qB(α)i − (A(qα)i−1 + B(qα)i)

= qA(α)i − A(qα)i = si ∈ S

So if the balanced representation B(α) is a sequence of input letters and B(qα)

is the sequence of output letters, then B(α) and B(qα) are in relation ρ(Mq). The

transitions

si−1B(α)i,B(qα)i−→ si

all belong to Mq.

Finally, we will obtain a one-to-one correspondence between tile sets and sequen-

tial machines which translates the properties of tile sets to the properties of the


calculation of sequential machines. We define the set of Wang tiles associated to the

machine Mq by the quadruples (si−1, B(α)i, si, B(qα)i), for i ∈ Z with

w(si−1, B(α)i, si, B(qα)i) = si−1, e(si−1, B(α)i, si, B(qα)i) = si; and

n(si−1, B(α)i, si, B(qα)i) = B(α)i, e(si−1, B(α)i, si, B(qα)i) = B(qα)i.

So we picture the quadruple (si−1, B(α)i, si, B(qα)i) as the tile:

( )i

B

1is

is

( )i

B q

Thus, we obtain a tiling x of Z2 as follows:

1( )

iB ( )

iB

1( )

iB

1is2i

si

s1i

s

1 1( )

iB q

1( )

iB q

1 1( )

iB q

1

2is

1

1is

1

is

1

1is

1 2 1( )

iB q q

1 2( )

iB q q 1 2 1

( )i

B q q

2

2is 2

1is

2

is

2

1is

1 2 3 1( )

iB q q q

1 2 3( )

iB q q q

1 2 3 1( )

iB q q q

Figure 5.10. The tiling x : Z2 → T.

Culik [11] described a sequential machine M = M3 ∪M′12

as in Figure.5.11. He

proved that the associated tile set T13 is aperiodic in the sense that it admits no

periodic tilings.


0, 2

1,1

1,2 1, 2

0,1 0,1

1

2

1,0

1,0 '

1,10 ',0

2,1 2,1

0 ',0

0

02 1

Figure 5.11. The sequential machine M = M3 ∪M′12

.

1

2

2

1

1

2

1

0

1

1

2

0

0

1

1

2

0

2 1

'0

0 1

02

0

0

1 '0

'0

'0

'0

2

0 1

'0 2

1

'0

1

'0

0

1

2

1

'0

'0

1

2

1

2

1

2

1

21

2

1

2

1

Figure 5.12. Aperiodic tile set T13 corresponding to the sequential machine M .

Example 5.23: We can construct a textile system (E, F, p, q) such that the associ-

ated system of Wang tiles is isomorphic to T13:


1f

1u

2u

3u

2f

3f

4f

5f

6f

4u

5u

1g

2g

3g

4g5

g

6g

7g

:F

:E

v

1e

2e

3e

4e

Figure 5.13. The textile system (E, F, p, q).


1f

2f

3f

4f

5f

6f

p q

1e 3

e

2e

2e

2e

3e

1e

2e

2e

3e

1e

2e

1g

p q

1e

2e

4e

2e

2e

2e

2g

3g

4g

5g

6g

7g

2e

2e

3e

4e

1e

3e

4e

1e

Figure 5.14. The graph morphisms p, q of E, F .

1e

1u

3e

3u1f

T

2e

1u3

u 2fT

2e

2e

1u 2

u3f

T

3e

1u2

u4f

T

1e

2e

3u5f

T3

u6f

T

2e

4u

5u1g

T2g

T

4e

2u

2u

1e

4u

5u

2e

3gT

4gT

4u

5gT 6g

T

2e

5u 7g

T

5u

2e

5u 4

u

3e

2e

4u

4u

4u

1e

4e

5u

5u

4e

1e

3e

2e

1e2

e

2e

3e

Figure 5.15. The tile set T′13 corresponding to the textile system (E, F, p, q).

Since the tile set T′13 is isomorphic to that of T13, the associated system (T, k, n, s, e, w)

of Wang tiles is aperiodic.

6. ENTROPY

In this chapter, we will introduce the idea of entropy. In [52], Entropy is described

to measure the complexity of mappings. For shifts, entropy also measures their “in-

formation capacity” or ability to transmit messages. In an information sense, entropy

is a measure of unpredictability.

Let A be a finite set, and k ∈ N. Let AZkbe the space of all maps x : Zk → A.

We call the elements of A symbols, and the elements of AZkcan be regarded as

infinite k-dimensional blocks of symbols from A. For any m,n ∈ Zk, the shift map

σm : X → X is defined as σm(xn) = xm+n. Such a shift map σ is homeomorphism,

and σmσn = σnσm = σm+n for all m,n ∈ Zk. A configuration on S ⊆ Zk is a map

ρ : S → A.

Given X ⊆ AZk, a configuration ρ on S ⊆ Zk is allowed for X if there exists

x ∈ X such that xS = ρ. A configuration ρ on S occurs in x ∈ AZkif ρ = σm(x)|S

for some m ∈ Zk. Let x ∈ AZkand S ⊆ Zk, we say x|S is the restriction of x to S.

Recall the definition of a k-dimensional shift space: given an alphabet A, a k-

dimensional shift space over A is a subset X ⊆ AZkwhich is closed under σ and

closed as a subset of the topological space AZk. For a subset X ⊆ AZk

, and l ∈ Nk,

an l-block of X is a configuration on [0, l] which is allowed for X. We write Bl(X)

for the collection of all l-blocks of X.

Example 6.1: Let k = 2, and for all p ∈ N, let l(p) = (p, p) ∈ N2. When p = 1,

l(1) = (1, 1). Then [0, l(1)] = (0, 0), (1, 0), (0, 1), (1, 1).

6. Entropy 112

(0,0) (1,0)

(1,1)(0,1)

Figure 6.1. The configuration of [0, (1, 1)]

We define B(X) =⋃

l∈Nk

Bl(X). Topological entropy [41] measures the complexity

of shift spaces. It measures the growth rate of the number of possible central squares

in the shift space.

In this section, we will present elements x of a shift space as diagram of a form

where each B(i,j) ∈ Bl(X) for some fixed l. ∗ indicates the position of (0, 0) in the

grid (see the following configuration).

(0,0)B (1,0)

B (2,0)B

(3,0)B

(0,1)B

(1,1)B (2,1)

B(3,1)

B

(0,2)B

(1,2)B

(2,2)B

(3,2)B

Definition 6.2: The topological entropy [52] of a k-dimensional shift space X is

defined to be

h(X) = limp→∞

1

pklog |Bl(p)(X)|.

In this chapter, we consider the case k = 2.

6. Entropy 113

Proposition 6.3: Let (E, F, p, q) be a textile system such that p, q have unique path

lifting. Let Λ(E, F, p, q) be the 2-graph of Theorem 3.8. Let X(E, F, p, q) = (Λ∆, σ)

be the associated shift space. Then, the entropy h(X(E, F, p, q)) = 0.

In [62], Skalski and Zacharias compute the topological entropy for a rank-r sub-

shift X of finite type, and give the similar result. Lewin and Pask in [30] show that

the two-dimensional shift space associated to each 2-graph has zero entropy. In this

thesis, we will give a new proof using the relationship between dual-graphs and textile

systems discussed in Chapter 3. Before we prove this proposition, we need to prove

some lemmas.

Lemma 6.4: Let (E, F, p, q) be a textile system such that p, q have unique path lift-

ing. Let Λ = Λ(E, F, p, q) be the associated 2-graph and let (1, 1)Λ be the associated

dual graph. Then (Λ∆, σ) is conjugate to ((1, 1)Λ∆, σ′).

Proof. Fix x ∈ Λ∆. Fix (m,n) ∈ Z2 such that m ≤ n. Define φ(x)(m,n) =

x(m,n + (1, 1)) ∈ (1, 1)Λn−m. Then φ(x) : ∆ → (1, 1)Λ is a k-graph morphism.

Hence φ(x) ∈ (1, 1)Λ∆ and φ defines a map from Λ∆ to (1, 1)Λ∆. We want to show

that φ is a homeomorphism such that (φ σp)(x) = ((σ′)p φ)(x) for all p. Fix

m ≤ n ∈ Z2 and p ∈ Z2. Then

(φ σp)(x)(m,n) = φ(σp(x))(m,n)

= σp(x)(m,n + (1, 1))

= x(m + p, n + p + (1, 1))

= φ(x)(m + p, n + p)

= (σ′)p(φ(x))(m,n)

= ((σ′)p φ)(x)(m,n).

Thus, we have φ σp = (σ′)p φ. In order to prove that φ is homeomorphism, we

need to show that it is onto and one to one, and that φ and its inverse function φ−1

are continuous.

For surjectivity, let y ∈ (1, 1)Λ∆. For (m,n) ∈ Z2 such that m ≤ n, let x : ∆ → Λ

be the unique function such that

y(m,n) = x(m,n)α for some α ∈ Λ(1,1) for all m ≤ n ∈ Z2. (6.1)

6. Entropy 114

We claim that x is a graph morphism. Fix x(m,n) = α, x(n, p) = β and x(m, p) = γ.

Then we can factorize y(m,n) = αµ, y(n, p) = βν where d(µ) = d(ν) = (1, 1). Since

y(m,n) and y(n, p) are composable in (1, 1)Λ∆, β = µβ′ and y(m,n)y(n, p) = αµβ′ν.

Since y(m, p) = γν ′ where d(ν ′) = (1, 1), the factorization property forces ν = ν ′ and

γ = αµβ′. Then x(m, p)x(n, p) = αµβ′ = γ = x(m, p). So x is a graph morphism.

We show that φ(x) = y. For m ≤ n ∈ Z2. Factorize y(m,n) = αµ where

d(α) = n −m and d(µ) = (1, 1). By (6.1), φ(x)(m,n) = x(m,n + (1, 1)) is defined

by y(m,n + (1, 1)) = φ(x)(m,n)α for some α ∈ Λ(1,1). So the factorization property

implies that y(m,n) = φ(x)(m,n) = x(m,n + (1, 1)). Thus, φ is surjective.

Next, we show φ is injective. Let x1, x2 ∈ Λ∆. Suppose that φ(x1) = φ(x2). Let

m ≤ n ∈ Z2. Then

x1(m,n)x1(n, n + (1, 1)) = x1(m,n + (1, 1)) = φ(x1)(m,n)

= φ(x2)(m,n) = x2(m,n + (1, 1)) = x2(m,n)x1(n, n + (1, 1)).

So the factorization property implies that x1(m,n) = x2(m,n). Thus, φ is injective.

To show φ and φ−1 are continuous, fix a basic open set Λ∆ as Z(λ, n) = x ∈ Λ∆ :

x(n, n+d(λ)) = λ of Λ∆. We claim that φ(ZΛ(λ, n)) =⋃

β∈Λ(1,1)

r(β)=s(λ)

Z(1,1)Λ(λβ, n). For the

“ ⊆ ” inclusion, fix x ∈ Z(λ, n). Then φ(x)(n, n+d(λ)) = x(n, n+d(λ)+(1, 1)) = αβ

where α ∈ Λd(λ) and β ∈ Λ(1,1). Then α = x(n, n + d(λ)) = λ, and hence φ(x) ∈Z(1,1)Λ(λβ, n) for some β ∈ Λ(1,1) with r(β) = s(λ). For the “ ⊇ ” inclusion, fix y ∈

⋃β∈Λ(1,1)

r(β)=s(λ)

Z(1,1)Λ(λβ, n). Since φ is surjective, there exists x ∈ Λ∆ such that φ(x) = y, and

then (6.1) implies that x ∈ ZΛ(λ, n). This proves the claim. Hence φ−1 is continuous.

Applying φ−1 to both sides of the claim gives ZΛ(λ, n) = φ−1(⋃

β∈Λ(1,1)

r(β)=s(λ)

Z(1,1)Λ(λβ, n)).

So φ is continuous as well.

From above, (Λ∆, σ) is conjugate to ((1, 1)Λ∆, σ′). 2

Proof. [Proof of Proposition 6.3] By Lemma 6.4, (Λ∆, σ) is conjugate to ((1, 1)Λ∆, σ′).

[31, Corollary 4.1.10] then implies that h(σ) = h(σ′). So we can replace Λ =

Λ(E, F, p, q) by (1, 1)Λ. The morphism φ : Bl(p)((1, 1)Λ) → (1, 1)Λl(p) is bijective.

Let A = Λ(1,1). Fix l(p) = (p, p) ∈ N2. Fix λ ∈ (1, 1)Λl(p) = Λ(p+1,p+1). For

λ ∈ (1, 1)Λl(p) and 0 ≤ i ≤ 2p, let a(λ)i ∈ A = Λ(1,1) be the unique element such that

6. Entropy 115

λ = µa(λ)iν in Λ where

d(µ) =

(i, 0), if i ≤ p;(p, i− p), if i > p.

The factorization property implies that λ 7→ (a(λ)0 . . . a(λ)2p) is an injection from

(1, 1)Λl(p) to (Λ(1,1))2p+1. Then, |Bl(p)((1, 1)Λ)| ≤ |A|2p+1. Let X := X(E, F, p, q).

Thus, the entropy of the shift space X = (Λ∆, σ) is

h(X) = limp→∞

1

pklog|Bl(p)(X)|

= limp→∞

1

p2log|Bl(p)((1, 1)Λ))|

≤ limp→∞

1

p2log|A|2p+1

= limp→∞

1

p2(2p + 1)log|A|

= 0. 2

We now show that for a general textile system (E, F, p, q), the entropy of the

associated shift space (X(E, F, p, q), σ) can be nonzero.

Definition 6.5: Let X = x : Z2 → T be a shift space. Let Br(X) = x|[0,r] : x ∈X ⊆ T [0,r]. The definition of topological entropy of a shift space X becomes

h(X) = limr→∞

1

r2log |Br(X)|.

We construct examples with nonzero entropy.

Example 6.6: Let E, F be the directed graphs.

u v

1b

1a:F

we f:E

2a

2b

1c

2c

1d

2d

6. Entropy 116

Define p, q : F → E by

p(a1) = p(b1) = p(c1) = p(d1) = e, p(a2) = p(b2) = p(c2) = p(d2) = f ;

q(a1) = q(b1) = q(c1) = q(d1) = f, q(a2) = q(b2) = q(c2) = q(d2) = e.

Define S := Ta1 , Ta2 , Tb1 , Tb2 , Tc1 , Tc2 , Td1 , Td2 be the tile set of the textile system

(E, F, p, q). We calculate the entropy of this textile system. Each 2-block B2(XT ) has

the form

(x2,1 x2,2

x1,1 x1,2

), and for all i, j ∈ 1, 2, xi,j are elements from S. Then

there are 8 possible values for x1,1 corresponding to the 8 edges in F ; once x1,1 is

fixed, there are 4 possible values for x1,2 corresponding to the edges of F 1 which are

composable with x1,1; then for x2,1, there are 4 possible values corresponding to the

edges in q−1 p(x1,1); finally, there are 2 possible values for x2,2 corresponding to the

edges in q−1 p(x1,2) that are compoable with x2,1. For example, if x1,1 = a1, then

x1,2 ∈ a1, a2, d1, d2, and x2,1 ∈ a2, b2, c2, d2; and if say x1,2 = d1 and x2,1 = c2,

then x2,2 ∈ b2, c2. Then |B2(XT )| = 8× 4× 4× 2. Next when r = 3, once B2(XT )

is fixed, for the 3-block B3(XT ), what we need to do is to find x1,3, x2,3, x3,3, x3,2, x3,1.

There are 4 possible values for x1,3 and x3,1, and 2 possible values for each x2,3, x3,2

and x3,3. Then, |B3(XT )| = |B2(XT )| × 4 × 4 × 2 × 2 × 2. Inductively, for general

r ∈ N, once Br(XT ) are fixed, there are 4 possible values for x1,r+1 and xr+1,1, and 2

possible values for x2,r+1, . . . , xr,r+1, xr+1,2, . . . , xr+1,r and xr+1,r+1. Then

|Br+1(XT )| = |Br(XT )| × 4× 2r−1 × 4× 2r−1 × 2

= |Br(XT )| × 42 × 22r−1

= (|Br−1(XT )| × 42 × 2r−2 × 2r−2 × 2)× 42 × 22r−1

. . .

= |B2(XT )| × (42)r−1 × 22r−1 × 22r−3 × . . . 21

= |B2(XT )| × (42)r−1 × 2r2

= |B2(XT )| × 2r2+4(r−1).

Then the entropy of shift space (XT , σ) is

h(XT ) = limr→∞

1

(r + 1)2log|Br+1(XT )|

6. Entropy 117

= limr→∞

1

(r + 1)2log(|B2(XT )| × 2r2+4(r−1))

= limr→∞

1

(r + 1)2log|B2(XT )|+ lim

r→∞r2 + 4(r − 1)

(r + 1)2log2

= log2 > 0.

We now formulate some conditions under which the shift space associated to a

given textile system has positive entropy.

Lemma 6.7: Suppose that (E, F, p, q) is a textile system such that p, q have path

lifting. Let (T, K, n, e, s, w) be the associated system of Wang tiles. Suppose there

exist cycles α1, . . . , αb of length a in F such that q(αj+1) = p(αj) for all 0 < j ≤ b− 1

and q(α1) = p(αb). Write αj := α(j,1), . . . , α(j,a) where α(j,i) ∈ F 1 for 0 < i ≤ a.

Suppose that there exists 0 < k ≤ a and edges β1, . . . , βb such that for each j ≤ b at

least one βj 6= α(j,k) and

(1) q(βj+1) = p(βj);

(2). p(βb) = q(β1) = q(α(1,k)) = p(α(b,k));

(3). r(βj) = r(α(j,k)), s(βj) = s(α(j,k)).

Then, h(XT ) ≥ 1ab

log2.

Proof. Define Cα, Cβ : [0, (a, b)] → T by Cα(i,j)= Tα(j,i)

and

Cβ(i,j)=

Tα(j,i)

, if i 6= k;

Tβ(j,i), if i = k.

For each r ∈ N, and each function Σ : 0, . . . , b rac × 0, . . . , b r

bc → α, β, the

formula

ρΣ(ma+i,nb+j) = CΣ(m+1,n+1)(i,j) where m < b rac, n < b r

bc

and 1 ≤ i ≤ a− 1, 1 ≤ j ≤ a− 1

defines a configuration on [0, (ab rac, bb r

bc)]. Fix r > 0. For each Σ as above, fix a

configuration ρΣ such that ρ|[0,(ab rac,bb r

bc)] agrees with ρΣ. Since at least one βj 6= αj,k,

we have Cα 6= Cβ. So if Σ 6= Σ′, then ρΣ 6= ρ′Σ.

We calculate a lower bound of the entropy of the shift space XT . We have

h(XT ) = limr→∞

1

r2log|Br(XT )| ≥ lim

r→∞log2b

racb r

bc

r2.

6. Entropy 118

Let r = ad1 + e1 and r = bd2 + e2, and di, ei ∈ Z+ ∪ 0 for i = 1, 2. Then

bracbr

bc = d1d2 ≤ d1d2 +

e1d2

a+

e2d1

b+

e1e2

ab

< d1d2 + (d1 + d2) + 1

≤ d1d2 + 2r + 1.

Hence,

limr→∞

b racb r

bc

r2≤ lim

r→∞

ra· r

b

r2=

1

aband

1

ab= lim

r→∞

ra· r

b

r2≤ lim

r→∞d1d2 + e1d2

a+ e2d1

b+ e1e2

ab

r2

< limr→∞

d1d2 + (d1 + d2) + 1

r2

= limr→∞

d1d2

r2+ lim

r→∞2r + 1

r2

= limr→∞

d1

r· d2

r

= limr→∞

b racb r

bc

r2.

Thus,

h(XT ) ≥ limr→∞

log2bracb r

bc

r2

= limr→∞

b racb r

bc

r2log2

= 1ab

log2 > 0. 2

In Lemma 6.7, a, b can be large positive integers. We show that h(XT ) can be an

arbitrarily small nonzero value by constructing a family of textile systems (E, F, p, q)

for which we can calculate the entropy of the associated shift space exactly.

Example 6.8: Fix m,n ≥ 0. Consider the following textile system (E, F, p, q), where

F has n connected components and each αi and each µi is a path of length m.

6. Entropy 119

1

1

1'

2

2

2'

...

n

n

'n

1...

...

2

n

1

2

n

:F :E

Figure 6.2. The associated shift space of (E, F, p, q) with positive of entropy.

This system clearly satisfies the hypotheses of Lemma 6.7 with b = m + 1 and

a = n. Let (E, F, p, q) be a textile system such that |αi| = |αj| = |µi| = |µj| for all

1 ≤ i ≤ j ≤ n and |βi| = |β′i| = |νi| for all 1 ≤ i ≤ n. Define the morphisms p, q as:

p(αi) = q(αi+1) = µi+1 for all 1 ≤ i ≤ n− 1, and q(α1) = p(αn) = µ1;

p(βi) = q(βi+1) = νi for all 1 ≤ i ≤ n− 1, and q(β1) = p(βn) = νn;

p(β′i) = q(β′i+1) = νi+1 for all 1 ≤ i ≤ n−1, q(β′1) = p(β′n−1) = νn, p(β′n) = q(β′n) = νn.

Remark 6.9: Let (E, F, p, q) be the textile system of Example 6.8. Let X(E, F, p, q)

be the associated shift space and let g = (|αj| + 1, n) ∈ N2. There are 2 special

distinct g-blocks B0, B1 of X(E, F, p, q) as follows:

B0(i, j) =

T(αj)i

, if i ≤ |αj|;Tβj

, if i = |αj|+ 1B1(i, j) =

T(αj)i

, if i ≤ |αj|;Tβ′j , if i = |αj|+ 1

Pictorially (see the following picture), where the entries of the left-hand columns

represent blocks of |αj| contiguous tiles,

n

11

... ...

n

0B

'n

1'

1

... ...

n

1B

6. Entropy 120

Lemma 6.10: Let (E, F, p, q) be a textile system as in Example 6.8 and let g be as

defined in Remark 6.9. The only g-blocks C satisfying C(0, 0) = T(α1)1 are C = B0

or C = B1.

Proof. Let C be a g-block with C(0, 0) = T(α1)1 . Then C(0, 0) . . . C(|α1|, 0) =

Tµ1 . . . Tµ|α1|+1for some µ ∈ F |α1|+1 with µ1 = (α1)1. Since µ = α1β1 and µ = α1β

′1

are the only such paths, then

C(0, 0) . . . C(|α1|, 0) ∈ T(α1)1 . . . T(α1)|α1|Tβ1 , T(α1)1 . . . T(α1)|α1|

Tβ′1.

Since q−1(p(αjβj)) = αj+1βj+1 and q−1(p(αjβ′j)) = αj+1β

′j+1 for all j ≤ n − 1,

then inductively,

C(0, j) . . . C(|αj|, j) =

T(α1)1 . . . T(α1)|α1|

Tβ1 , if µ = α1β1;

T(α1)1 . . . T(α1)|α1|Tβ′1 , if µ = α1β

′1.

Thus, C = B0 or C = B1. 2

Corollary 6.11: With the hypothesis of Lemma 6.10, if x ∈ X(E, F, p, q) and

x|[0,g] ∈ B0, B1, then σ(g1,0)(x)|[0,g], σ(−g1,0)(x)|[0,g], σ

(0,g2)(x)|[0,g] and σ(0,−g2)(x)|[0,g]

belong to B0, B1.

Proof. To obtain this conclusion, by Lemma 6.10 it suffices to show that

σ(g1,0)(x)(0, 0) = σ(−g1,0)(x)(0, 0) = σ(0,g2)(x)(0, 0) = σ(0,−g2)(x)(0, 0) = T(α1)1 .

Since x|[0,g] ∈ B0, B1, then

x(0, 0) . . . x(|α1|, 0) ∈ T(α1)1 . . . T(α1)|α1|Tβ1 , T(α1)1 . . . T(α1)|α1|

Tβ′1.

Hence, x(|α1|, 0) ∈ Tβ1 , Tβ′1. Since e(Tβ1) = e(Tβ′1) = s(β1) = s(β′1), and since

r−1(s(β1)) = r−1(s(β′1)) = (α1)1, we deduce that x(|α1| + 1, 0) = T(α1)1 . Hence,

σ(g1,0)(x)(0, 0) = T(α1)1 . Similarly, since w(x(0, 0)) = w(T(α1)1) = r((α1)1), and since

s−1(r((α1)1)) ∈ β1, β′1, we have x(−1, 0) ∈ Tβ1 , Tβ′1. Since s−1(r(β1)), s

−1(r(β′1)) ∈(α1)|α1|, and since s−1(r(α1)i+1)) = (α1)i for all 1 ≤ i ≤ |α1| − 1, then

x(−|α1| − 1, 0) . . . x(−1, 0) ∈ T(α1)1 . . . T(α1)|α1|Tβ1 , T(α1)1 . . . T(α1)|α1|

Tβ′1.

Hence, x(−|α1| − 1, 0) = T(α1)1 , and so σ(−g1,0)(x)(0, 0) = T(α1)1 .

6. Entropy 121

Now, consider σ(0,g2)(x)(0, 0) and σ(0,−g2)(x)(0, 0). In the case of σ(0,g2)(x)(0, 0),

Since q−1(p(αjβj)) = αj+1βj+1 and q−1(p(αjβ′j)) = αj+1β

′j+1 for all j ≤ n − 1,

then x(0, n − 1) = T(αn)1 . Moreover, q−1(p(αn)) = α1. Hence, x(0, n) = T(α1)1 .

Thus, σ(0,g2)(x)(0, 0) = T(α1)1 . Similarly, since p−1(q(α1)) = αn, x(0,−1) = T(αn)1 .

Moreover, since p−1(q(αjβj)) = αj−1βj−1 and p−1(q(αjβ′j)) = αj−1β

′j−1 for all

−n ≤ j ≤ −2, we have x(0,−n) = T(α1)1 . Thus, σ(0,−g2)(x)(0, 0) = T(α1)1 . ¤

Lemma 6.12: Let (E, F, p, q) be a textile system as in Example 6.8 and let g be as

defined in Remark 6.9. Let x ∈ X(E, F, p, q). Then there exist unique k ≤ |αj| + 1

and l ≤ n such that σ(k,l)(x)|[0,g] ∈ B0, B1.

Proof. Firstly, we define k and l. If x(0, 0) = T(αj)i, define k = (2− i)(mod |αj|+1)

and l = (1− j)(mod n). If x(0, 0) = Tβjor Tβ′j , define k = 1 and l = (1− j)(mod n).

To see that σ(k,l)(x)|[0,g] ∈ B0, B1 it is suffices by Lemma 6.10 to show that

σ(k,l)(x)(0, 0) = T(α1)1 . We consider two cases.

Case 1: Suppose x(0, 0) = T(αj)ifor some j ≤ n and i ≤ |αj|.

Since r−1(s((αj)i)) = (αj)i+1 for all j ≤ n and i ≤ |αj| − 1, then we have

x(1, 0) = T(αj)i+1. Continuing in this way, we have x(|αj| − i, 0) = T(αj)|αj |

. Since

r−1(s((αj)|αj |)) = βj, β′j, we have x(|αj| − i + 1, 0) ∈ Tβj

, Tβ′j. Since r−1(s(βj)) =

r−1(s(β′j)) = (αj)1, then x(|αj| − i + 2, 0) = T(αj)1 . The only paths µ in F |αj |+1

with µ1 = (αj)i are:

(1) (αj)i(αj)i+1 . . . (αj)|αj |βj(αj)1 . . . (αj)i−1; and

(2) (αj)i(αj)i+1 . . . (αj)|αj |β′j(αj)1 . . . (αj)i−1.

Since x(0, 0)x(1, 0) . . . x(|αj|, 0) = Tµ1Tµ2 . . . Tµ|αj |+1for some such µ, we have

x(k, 0) = T(αj)1 . Since q−1(p((αj)1)) = (αj+1)1 for all j ≤ n − 1, it follows that

x(k, 1) = T(αj+1)1 . Continuing in this way, we have x(k, n − j) = T(αn)1 . Since

q−1(p((αn)1) = (α1)1, we have x(k, l) = x(|αj| − i + 1, n− j + 1) = T(α1)1 .

Case 2: Suppose x(0, 0) = Tβjor Tβ′j .

With similar argument of case 1, x(1, 0) = T(αj)1 , and the proof proceeds as above

to give x(1, n− j + 1) = T(α1)1 . 2

6. Entropy 122

( )j i

(0,0) (1,0) ( 2,0)i

1( )

j

( 1,0)i

/ 'j j

1( )

j i

...... ( )j

( ,0)i

......(0,1)

1( )

j i

(1,1)

1 1( )

j i

( 2,1)i

1( )

j

( 1,1)i

1 1/ '

j j

( ,1)i

1 1( )

j

....................................

......(0, )n j

( )n i

(1, )n j

1( )

n i

( )n

/ 'n n

( , )i n j

1( )

n

......1

( )i

(0, 1)n j

1 1( )

i 1( ) 1 1

/ '( , 1)i n j

1 1( )

(1, 1)n j

Figure 6.3. The configuration of σ(k,l)(x)(0, 0) = T(α1)1 .

Corollary 6.13: Let (E, F, p, q) be a textile system as in Example 6.8. Let g be as

defined in Remark 6.9 and let k, l be as in the statement of Lemma 6.12. Then there

exists L : Z2 → 0, 1 such that for any p, q ⊆ Z,

σ(k,l)(x)|[(p(|αj |+1),qn),((p+1)(|αj |+1),(q+1)n)] ≡ BL(p,q). (6.2)

Pictorially,

(0,0)LB

(1,0)LB

(2,0)LB

( 1,0)LB

( 1,1)LB

(0,1)LB

(1,1)LB (2,1)L

B

(2, 1)LB

(1, 1)LB

(0, 1)LB

( 1, 1)LB

( , )( )

k l

x

where the block BL(0,0) in the picture has its bottom left corner at the origin. More-

over, for any L : Z2 → 0, 1, the formula (6.2) defines an element xL ∈ X(E, F, p, q).

Proof. From Lemma 6.12, there exist k, l such that σ(k,l)(x)|[0,g] ∈ B0, B1. Define

L(0, 0) to be the element of 0, 1 such that σ(k,l)(x)|[0,g] = BL(0,0). We now establish

(6.2) by induction using Corollary 6.11.

6. Entropy 123

We have r(αj) = s(βj) = s(β′j) and s(αj) = r(βj) = r(β′j) for all j. For the final

statement, observe that w(Bp(1, k)) = e(Bq(|αj| + 1, k)) for all k ≤ n and all p, q ∈0, 1, so the formula (6.2) satisfies e–w condition. Similarly, since n(Bp(h, n)) =

s(Bq(h, 1)) for all h ≤ |αj| + 1 and p, q ∈ 0, 1, the formula (6.2) satisfies n–s

condition. Hence, (6.2) gives a valid tiling. 2

Theorem 6.14: For the textile system of Example 6.8, we have h(X(E, F, p, q)) =

1(|α1|+1)n

log2.

Proof. Let X := X(E, F, p, q). Lemma 6.7 implies that h(X) ≥ 1(|α1|+1)n

log2. Fix

r ∈ N. We must estimate |Br(X)|.We claim that |Br(X)| ≤ 2

d r|α1|+1

ed rne · (|α1|+ 1) ·n. To see this, it suffices to show

that for x ∈ X:

x|[0,(r,r)] ∈ xL′|[(k,l),(k+r,l+r)] : k ≤ |α1|+ 1, l ≤ n and L′(a, b) = 0 unless

(0, 0) ≤ (a, b) ≤ (d r|α1|+1

e, d rne). (6.3)

Fix x ∈ X. Define L′ : Z2 → 0, 1 by L′|[(0,0),(d r|α1|+1

e,d rne) = L|[(0,0),(d r

|α1|+1e,d r

ne)],

L′(a, b) = 0 for (a, b) /∈ [(0, 0), (d r|α1|+1

e, d rne)]. Then

xL′|[(0,0),(d r|α1|+1

e,d rne) = xL|[(0,0),(d r

|α1|+1e,d r

ne) = σ(k,l)x|[0,(r,r)].

Now, x|[0,(r,r)] = σ(−k,−l)(x)|[(k,l),(k+r,l+r)] = xL′|[(k,l),(k+r,l+r)], which gives (6.3).

(0,0)x

( , )x r r

( , )x k l

Thus,

|Br(X)| = |x(0, (r, r)) : x ∈ X|

6. Entropy 124

≤ 2d r|α1|+1

ed rne · (|α1|+ 1) · n.

Hence,

h(X) ≤ limr→∞

log2d r|α1|+1

ed rne · (|α1|+ 1) · nr2

= limr→∞

log2d r|α1|+1

ed rne

r2+ lim

r→∞log(|α1|+ 1) · n

r2.

We have d r|α1|+1

ed rne ≤ ( r

|α1|+1+ 1)( r

n+ 1), and (|α1| + 1) · n is fixed. Let M =

(|α1|+ 1) · n. Then

h(X) ≤ limr→∞

log2( r|α1|+1

+1)( rn

+1)

r2+ lim

r→∞logM

r2

= limr→∞

( r|α1|+1

+ 1)( rn

+ 1)log2

r2+ 0

=1

(|α1|+ 1)nlog2.

Therefore, h(X) ≤ 1(|α1|+1)n

log2. From above, h(X) = 1(|α1|+1)n

log2, so we have

equality. 2

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