TOPOLOGICAL EFFICIENCY OF STIRRING WITH OBSTACLES

By

JASON C. HARRINGTON

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

c⃝ 2010 Jason C. Harrington

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To my wife and my family who supported me throughout the whole process

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ACKNOWLEDGMENTS

I am indebted to my PhD advisor, Professor Boyland, for his patience and guidance

throughout the whole PhD process. Without his help, this dissertation would not be

possible. I would like to express my gratitude to the committee members. I would like to

thank Professor Keesling and Professor Block for introducing me to dynamics and for

their excellent teaching. I would also like to express my gratitude to Professor Robinson

for his insightful comments on the dissertation and his general guidance while I was a

student in graduate school. Lastly, I would like to thank Professor Peters for serving as

the external committee member.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Stirring Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Stirring as a Braid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Thurston’s Analysis of Mapping Classes . . . . . . . . . . . . . . . . . . . 161.7 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 BURAU REPRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Burau Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.1 Burau as the Action on Homology . . . . . . . . . . . . . . . . . . . 192.1.2 Right Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Reduced Burau Representation . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Burau Representation and Entropy . . . . . . . . . . . . . . . . . . . . . . 242.4 Convention and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 FUNDAMENTAL STIRRING PROTOCOLS . . . . . . . . . . . . . . . . . . . . 26

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Configuration Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 NUMERICAL EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Burau Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Mousifir Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Results of Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 PROPERTIES OF HSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1 Entropy Lower Bound for HSP . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Gearability of HSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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6 UPPER BOUNDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1 Construction of Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Spectral Radius of incidence Matrices . . . . . . . . . . . . . . . . . . . . 51

APPENDIX: SOME GENERAL MATRIX LEMMAS . . . . . . . . . . . . . . . . . . . 59

A.1 General Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.2 Spectral Radius Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 60

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6

LIST OF FIGURES

Figure page

1-1 Particle Dance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1-2 Geometric Braid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1-3 Sigma Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2-1 π1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-2 σi Acting on π1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-3 Z cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2-4 Lift of a path in M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2-5 1-Skeleton of M2 and its lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2-6 Induced action of σ1 on Z-cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-1 α4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4-1 Two versions of HSP6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4-2 Comparison between two stirring protocols . . . . . . . . . . . . . . . . . . . . 31

5-1 Efficiency of HSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5-2 Train Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5-3 Constructing HSP(8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6-1 New Generators for π1(M2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6-2 New Alpha Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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Abstract of dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

TOPOLOGICAL EFFICIENCY OF STIRRING WITH OBSTACLES

By

Jason C. Harrington

December 2010

Chair: Phillip BoylandMajor: Mathematics

We investigate the two-dimensional problem of stirring by a single stirrer moving

around a finite number of obstacles. The goal is to find the stirring protocol which yields

the maximum topological efficiency. Topological efficiency is defined to be the entropy

associated with the stirring protocol divided by the number of stirrer motions around

each obstacle, and we denote the maximum efficiency with m obstacles as E(m).

Our investigation begins with numerical simulations of the topological efficiency

for a variety of paths and number of obstacles. Since we seek the maximum efficiency,

the topological efficiency of any specific stirring protocol yields a lower bound for E(m).

To the degree possible computationally we find that the maximum efficiency seems to

be realized by a specific protocol which is essentially “looping around each obstacle

once,” and we denote it HSPm. Using techniques from Thurston-Nielsen theory we can

rigorously compute the efficiency of HSPm, use it as a lower bound for E(m), and show it

converges to log(3) as m → ∞.

For an upper bound, we will use Bowen’s Theorem that entropy is bounded

below by the growth rate of the action of the map on the fundamental group. We use

non-standard generators for our space, the disk with m + 1 holes. The stirring path

induces an action on these generators. The growth rate on π1 is overestimated by

ignoring cancellations. This allows us to use the incidence matrices of the action and

thus get an entropy upper bound using the generalized spectral radius of the incidence

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matrices of the fundamental motions of the stirrer. We prove that this generalized

spectral radius is actually achieved by a finite product of incidence matrices which are

closely related to the protocol HSPm. Finally, we show that the upper bound for the

topological efficiency converges to log(3) as m → ∞, and so, E(m) is log(3) in the

asymptotic limit.

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CHAPTER 1INTRODUCTION

1.1 Introduction

We investigate the two-dimensional problem of stirring by a single stirrer moving

around a finite number of obstacles. We call such a stirring motion as a fundamental

stirring protocol. The goal is to find the fundamental stirring protocol which yields the

maximum topological efficiency. Topological efficiency is defined to be the entropy

associated with the stirring protocol divided by the number of stirrer motions around

each obstacle, and we denote the maximum efficiency with m obstacles as E(m). In

other words,

E(m) = sup{E� (β) : β is a fundamental stirring protocol with n − 1 fixed holes }

where E� denotes the topological efficiency of the fundamental stirring protocol.

Theorem 1.1 (Main Theorem). Consider a disk with m − 1 obstacles, and one funda-

mental stirrer. Then

log((3m−1 − 3m + 2)/(m − 1))

m − 1≤ E(m) ≤ log(3).

We begin by looking at two-dimensional stirring from different points of view. First,

we will consider a stirring protocol as a particle dance, where the stirrers move around

as points in two-dimensions. This will describe the possible paths of the stirrers. Next,

we consider a stirring protocol as a braid. Since braids have generators, this enables us

to have a group structure on stirring protocols. We will use αj ’s to denote the elements

of a fundamental stirring protocol.

Next, we will consider a stirring protocol as an element of the mapping class group,

which enables us to use Thurston-Nielsen theory. When a stirrer moves around an

obstacle, it will “pull” and “stretch” the fluid around. However, given a specific stirring

protocol, different fluids may behave differently. For example, stirring coffee with cream

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is rather effortless and can be done with a flick of a spoon, whereas this motion will

not mix thicker fluids like cement. If we consider a fluid as a surface homeomorphism,

then Thurston-Nielsen theory gives us the a surface homeomorphism called φ that

has the smallest entropy associated with the stirring protocol. We will use the Burau

representation for braids which gives a lower bound for the entropy of φ.

Since any stirring protocol gives a lower bound for E(m), we used several numerical

techniques for finding one that appeared to be most efficient and then rigorously

computed its efficiency. Three different different numerical techniques for approximating

the entropy of a fundamental stirring protocol with n − 1 fixed holes were used. Several

thousand different stirring protocols were generated and tested with these techniques.

1. The first technique uses the Burau representation of the fundamental stirringprotocol. The matrix generated at t = −1 gives a lower bound for the entropy ofthe braid.

2. The next technique considers the action of the fundamental stirring protocol on thefundamental group of the disk with n holes by allowing the fundamental stirrer tobe one the holes. The stirring protocol was then iterated up to 10 times. This wasused to approximate the exponential growth on π1 and thus the entropy.

3. The last technique uses a program called trains.exe created by Toby Hall thatuses train tracks to compute the entropy of a braid. A random fundamental stirringprotocol was generated and then converted into the generators for the Artin braidgroup and inputed that word into the program.

After considering several thousand different stirring protocols that were produced

randomly, the one that was most efficient out of tested protocols was found to be a

protocol which loops exactly once around each obstacle in succession. This stirring

protocol was used to compute a lower bound.

In order to compute the upper bound, we used the action that is induced on the

fundamental group by the α generators for the fundamental stirring protocols. By not

considering cancellation, we were able to find an upper bound for the exponential growth

on π1 and thus the entropy.

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Putting both of these ideas together, we are able to show that

log((3m−1 − 3m + 2)/(m − 1))

m − 1≤ log(ρ(HSPm))

m − 1≤ E(m) ≤ log(3).

This means that limm→∞ E(m) = log(3) and that limm→∞ log(ρ(HSPm))/(m − 1) = log(3).

The process of computing the upper bound produced a matrix, named the incidence

matrix, for each generator of the fundamental stirring protocol. Let Aj denote the

incidence matrix for αj . The generalized spectral radius of the set of incidence matrices

is computed. The generalized spectral radius of the set of matrices is realized by the

matrix A2 ...An which is exactly the same order as in the lower bound. This supports the

conjecture that E(n) is actually achieved by the HSP.

1.2 Stirring Protocol

We now discuss the set-up to study fluid mixing from a topological point of view.

This branch of mathematics is called Topological Fluid Mechanics. First we will define

what is meant by a general stirring protocol. Let D2 be the unit disk in R2 and let n be a

positive integer.

Definition 1 (Stirring Protocol). Consider n distinct points x1, x2, ... , xn in the interior of

D2. We define a stirring protocol SP by SP = (f1(t), ... , fn(t)) where fi : [0, 1] → D2 such

that:

1. fi is continuous for each i ,2. fi(t) = fj(t) if i = j , for all t ∈ [0, 1],3. fi(0) = xi for each i ,4. fi(1) ∈ {x1, x2, ... , xn} for each i .

Each fi corresponds to the motion of a stirrer. Some people prefer that the path of

the stirrer be differentiable, however, differentiability is not needed for this discussion.

Condition 2 implies that the stirrers do not hit each other at each time t, i.e. {f1(t), ... , fn(t)}

are n distinct points. Condition 3 merely tells us the starting position of the stirring

protocol. Condition 4 implies the stirrers must end back on the set {x1, x2, ... , xn}.

This definition of a stirring protocol is sometimes referred to as a ”particle dance” as

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discussed by Rolfsen [1]. Figure 1-1 gives an example of a stirring protocol, where

there are initially four stirrers and the arrow shows the path of each stirrer back to

the ”base” position. Note that it is possible for fi(1) = fi(0) for some i . If we replace

Figure 1-1. The stirrers act as particles moving in the fluid.

condition 4 with “fi(1) = fi(0) for all i ” then we have a special case called a pure stirring

protocol.

1.3 Stirring as a Braid

We will now show that the construction of a stirring protocol can be described in

terms of the Artin Braid group. The rigorous proofs of the claims in the section can be

found in the classic Birman book [2].

Let SP be a stirring protocol as defined above. Let I = [0, 1] and Ai : I → I × D2

be defined by Ai(t) = (t, fi(t)) for each i . By construction, each Ai is a continuous arc.

We will call each Ai a braid string. Let A = {A1,A2, ... ,An} be a family of arcs that start

at the points {(0, x1), (0, x2), ... , (0, xn)} and end up at {(1, f1(1)), (1, f2(1)), ... (1, fn(1))}.

By construction, each Ai intersects each intermediate parallel plane exactly once, and

A intersects each intermediate parallel plane at exactly n distinct points. We call A a

geometric n-braid.

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Definition 2 (Pure Braid). If all the arcs of a geometric n-braid, A, have the property that

fk(1) = xk for all k , where f is a stirring protocol and xk is the starting point as defined

previously, then we call this braid a pure braid.

Figure 1-2. A stirring protocol as a Geometric Braid.

Next, we will discuss the equivalence of two geometric n-braids. The idea is that

we should be able to “deform a braid” without changing the structure, i.e. a braid string

can not pass through other braid strings but can “wiggle” around. We make this concept

more precise by the following definition.

Definition 3 (Equivalence of Geometric Braids). Let X = I × D2. We say that two

geometric braids β and β′ are equivalent if there is an isotopic deformation Hs : X × I →

X where the image set βs is a geometric braid for all s ∈ I and β0 = β and β1 = β′.

Let B(n) denote the set of all equivalence classes of the geometric n-braids. Let

β1, β2 ∈ B(n), we define the product (composition), denoted as β1 · β2 as follows.

Let β1 = [B1] where [B1] denotes the equivalence class for the geometric braid B1,

similarly, β2 = [B2]. So B2 is a family of arcs that start at {(0, x1), (0, x2), ... , (0, xn)} and

end at {(1, x1), (1, x2), ... , (1, xn)}. Shift B1 up so that B1 is a family of arcs that start at

{(1, x1), (1, x2), ... , (1, xn)} and end at {(2, x1), (2, x2), ... , (2, xn)}. We concatenate the

braids together with B1 on top and B2 below. Together, we have a braid in [0, 2] × D2.

We may rescale time, so that the concatenated braid fits in I × D2. The equivalence

class of this newly formed braid is β1 · β2. There is also an identity element, namely, the

braid where each string is simply a straight line and does not cross other strings. Given

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a braid β1, there exists a braid β3 that will “undo” β1 in the sense that β1 · β3 is the identity

braid.

The Artin Braid group is the group formed from (B(n), ·). Artin identified a set of

generators and relations, i.e. a presentation for the Artin Braid group. Let σi denote the

braid whose i th and i + 1th braid string change places as shown in Figure 1-3. In Artin’s

Figure 1-3. σi and σ−1i

first paper on braid groups from 1925 [3], he gave a presentation of the braid group in

terms of the σi ’s and their relations, namely:

Theorem 1.2 (Artin). The group B(n) of geometric braids on n-strings admits a presen-

tation with generators σ1,σ2, ... ,σn−1 with the defining relations;

1. σi · σj = σj · σi for |i − j | > 2,2. σi · σi+1 · σi = σi+1 · σi · σi+1 for 1 ≤ i ≤ n − 2.

By convention, we will have time going down and will think of the σ’s as composition,

so we will read them from right to left. For example, from the braid in the figure 1-2

the braid presentation is σ2σ−11 . We will describe the stirring protocol in terms of the

generators of the Artin braid group.

1.4 Mapping Class Groups

We have another way we may describe a stirring protocol. The description will help

us understand the dynamics given by a stirring protocol.

Let M2 be an orientable, compact surface with negative Euler characteristic,

possibly with boundary. We will allow a finite number of points that act like obstacles.

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Definition 4. The mapping class group on M2 is

MCG(M2) = {[f ] where f : M2 → M2is a homeomorphism and

f |∂M2 = id and [f ] is the isotopy class for f }

where isotopies are taken to fix the boundary, ∂M2, point wise. MCG has a natural group

operation, namely composition.

Let M2 = D2 − {n − pts}. Imagine that the points removed are stirrers in a “fluid”

and the points pull their surroundings with them as they move about. Topologically, the

motion of the points extends (not uniquely) to a continuous family of homeomorphisms

of M2. This isotopy class describes an equivalence between B(n) and MCG(M2).

1.5 Topological Entropy

Let γ : M2 → M2 where γ is a homeomorphism and γ∗ : π1(M2) → π1(M

2) be the

group action induced by γ. In other words, as γ moves the points of the space around,

then γ∗ will also move the elements of the fundamental group in M2 in the same manner.

There is a nice geometrical description in figure 2-2 when γ = σi .

Let L denote the word length. By Bowen [4] with a theorem in [5] the topological

entropy of a pseudo-Anosov map γ is given by

h(γ) = supg∈π1(M2)

limm→∞

log(L (γm∗ g))

m.

Entropy can be used to determine the asymptotic behavior of iterating an induced

action. For example, if h(γ) > 0, then we know that the word length of γm∗ g is growing

exponentially for all g ∈ π1(M2) as m increases. Furthermore, if h(γ1) > h(γ2) for some

induced actions on π1(M2), then (γ1)

m∗ g is growing exponentially faster than (γ2)

m∗ g for

all g as m increases This gives a measure of the complexity of the action.

1.6 Thurston’s Analysis of Mapping Classes

We would like to discuss the dynamics of the stirring protocol. We will make use of

the following theorem:

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Theorem 1.3 (Thurston-Nielsen Classification). If f is a homeomorphism of a compact

surface, S , with perhaps a finite number of punctures, then f is isotopic to a homeomor-

phism, φ, of one of the following types:

1. Finite order: φm = id for some integer m ≥ 1;

2. Pseudo-Anosov: φ preserves a pair of transverse, measured foliations, Fu and Fs

and there is a λ > 1 such that φ stretches Fu by a factor of λ and contracts Fs by1/λ;

3. Reducible: φ fixes a family of reducing curves, and on the complementary surfaceφ satisfies (1) or (2).

This theory was developed by Thurston [6]. Letting f ∈ MCG(M2), we have three

cases for the Thurston representative φf . The finite order case states that after a finite

number of iterations of ϕf , we will end up back where we started. The pseudo-Anosov

case, abbreviated pA, tells us that the topological entropy of φf is log(λ) where λ is

the value stated in the theorem [5]. The third case, the reducible case, implies that the

space can be divided into invariant pieces where φf has either finite order or is pA on

the pieces.

1.7 Fluid Mechanics

In this section, we will explain how a stirring protocol relates to the physical

application of fluid mixing. By the following theorem of Handel [7], we see that a pA

map gives us a lower bound for the dynamics induced by a stirring protocol.

Theorem 1.4 (Handel’s Istopy Stability). If φ is a pA and f is isotopic to φ, then there

is a compact, f -invariant set, Y , and a continuous, onto mapping α : Y → M2, so that

αf = φα.

Thus, we have the commutative diagram,

Y

�

f // Y

�

M2φ

// M2

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This theorem states that the dynamics of f are at least as complicated as the

dynamics of φ. In the case with fluid mixing, the fluid moves continuously on the surface

D2 −{n− pts}. Let f denote the fluid flow. So if we have a pA map called, φf , associated

to f , then the topological entropy log(λ) = h(φf ) ≤ h(f ) where λ is the value from

Thurston’s theorem. So the pA gives the lower bound for the topological entropy of any

fluid, regardless of its physical properties.

So with each braid, there is a Thurston representative. These representatives give

us a lower bound for the entropy of the actual fluid flow. The idea is to maximize the

amount of entropy associated with stirring, while minimizing the number of motions

needed for the stirrer to move.

Historically, fluid mixing was solved on a case-by-case basis. The fluid flow

would be a solution to a specific set of partial differential equations, and a computer

experiment was run with various stirring protocols. Then to determine the best stirring

protocol, one would use a specific mixing measure and compare. Finn, Cox and Burns

[8], list a few various measures that are still used today. Also, Ottino [9] and Chorin [10]

are good references for traditional fluid mechanics.

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CHAPTER 2BURAU REPRESENTATION

2.1 Burau Representation

A representation of a group G on a module M over a ring R is a group homomorphism

from G to GL(M). We define a homomorphism

b : B(n) → GLn(Z[t, t−1])

called the Burau representation, by the images of the generators of B(n).

b(σi) =

Ii−1 0 0 0

0 1− t t 0

0 1 0 0

0 0 0 In−i−1

i th row

where Ik denotes the k × k identity matrix. This representation is named after its

discoverer Burau in 1936 [11].

We see by direct computation that

1. b(σi) · b(σj) = b(σj) · b(σi) for |i − j | > 2,2. b(σi) · b(σi+1) · b(σi) = b(σi+1) · b(σi) · b(σi+1) for 1 ≤ i ≤ n − 2.

The Burau representation for n = 2, 3 is faithful and the proof can be found in

Birman’s book [2]. In 1999, Bigelow [12] showed that the representation is not faithful for

n = 5. For n ≥ 6, Long and Paton showed in 1993 that the Burau representation is not

faithful [13]. However, it is still unknown whether or not it is faithful when n = 4.

2.1.1 Burau as the Action on Homology

We will show that this representation can be interpreted as the induced action on

homology in Z-covers by a braid β. References for this section are [14] and [15]. First

we must introduce another action.

19

2.1.2 Right Automorphism

We need to compute the action of the braid generators on the fundamental group

of our disk minus n points, M2. Let Fn denote a free group on n symbols. Recall the

σ notation for the braid generators from 1.2. We will define a right automorphism of

Fn =< x1, ... , xn >. Let σi ∈ B(n) and we will define σ ∈ Aut(Fn) by

xi σi = xixi+1x−1i (2–1)

xi+1σi = xi (2–2)

xj σi = xj when j = i and j = i + 1. (2–3)

Recall that σi switches the braid i th hole. There is a graphical interpretation for this

action. Let xi be a generator for M2 = D2 − {n − pts}, i.e. We determine how the σi

x1 x2 x3 xn

p

Figure 2-1. π1 Generators

generators act on π1(M2, p). We think of σi as twisting the loops xi and xi+1. Notice that

the other loops, xj do not move, i.e. xj σi = xj for all j = i and j = i + 1.

x1 xi xi+1 xn

p

x1 xn

p

Figure 2-2. σi Acting on π1 Generators

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Now, we can construct a covering space ~M2 of M2 by making cuts from each

puncture to the boundary and connecting copies of M2. For example, when n = 3 we

have the following figure. Thus, the deck transformations mapping from one copy to

Figure 2-3. Cover of a disk with 3 holes

the next are given by the group < t >, the cyclic group of infinite order. This process of

passing from homotopy to homology is known as Abelianization.

For example, when n = 3 we have the following figure.

p

p

tp

Figure 2-4. Loop in M2 lifts to a path from ~p to t~p

21

By the homotopy lifting property, we know after fixing a base point in the cover that

a loop lifts uniquely to a path in the cover. We will now consider the action on ~M2 by

looking at a 1-skeleton of M2 and its lift, as we see below:

a

b

c

a b c~ ~ ~pp

tp

~

~

t

Figure 2-5. 1-Skeleton of M2 and its lift

Next, we consider a specific example, with σ1 we have that a 7→ aba−1 from 2–1

where a ∈ π1(M2) from 2-5 and a is in the lift of a. Let ~σ1 denote the lift of the action σ1

in the 1-skeleton. Thus, a~σ1 = (1 − t)a + b and similarly a~σ1 = a and c~σ1 = c . We

p

tp

~

~

a~

σ p

tp

~

~

a~

1

ta~ tb~

~

Figure 2-6. Induced action on Z-cover

can repeat the process to compute the action of the other generators. We will write the

22

action ~σi by a matrix. This is the action of ~σ1,

~x1 7→ ~x1 − t ~x1 + t ~x2,

~x2 7→ ~x1,

~x3 7→ ~x3.

We can write ~σ1 as matrix by observing that

~σ1 :

~x1

~x2

~x3

7→

1− t t 0

1 0 0

0 0 1

~x1

~x2

~x3

.

We have now arrived at the Burau representation for σ1, and similarly we can construct

matrix representation for the other generators.

From our example in which n = 3 and ~σ1 we may choose a different basis,

( ~x1 + t ~x2 + t2 ~x3, ~x2 − ~x1, ~x2 − ~x3)

This yields

b(σi)

~x1 + t ~x2 + t2 ~x3

~x2 − ~x1

~x2 − ~x3

=

1 0 0

0 −t 0

0 −1 1

x1 + tx2 + t2x3

x2 − x1

x2 − x3

The matrix in the block is called the reduced Burau representation. For a braid group

B(n), we only need to consider the reduced Burau representations in GLn−1(Z[t, t−1]) to

compute the lower bound for entropy.

2.2 Reduced Burau Representation

We define a homomorphism

rb : B(n) → GLn−1(Z[t, t−1])

23

called the reduced Burau representation, by the images of the generators of B(n).

rb(σ1) =

−t 1 ... 0

0 I

rb(σj) =

1

. . .

1 0 0

t −t 1

0 0 1

. . .

1

where (0, ... , 0, t,−t, 1, 0, ... , 0) occurs on the j th row for r(σj) and 1 < j < n.

2.3 Burau Representation and Entropy

Let MSP(t) be a reduced Burau matrix representation of a stirring protocol SP.

Let ρ(M) denote the spectral radius of M. Let h(SP) denote the entropy of the stirring

protocol. Fried [16] has shown that

Theorem 2.1 (Fried).

h(SP) ≥ sup{ρ(MSP(t)) : t ∈ C |t| = 1} (2–4)

as a corollary of results using twisted cohomology. Later Kolev in [17] showed this

inequality using only the definition of the Burau representation and Fox free differential

calculus. We will call equation 2–4 the Burau estimate. Then the question was asked,

“under what conditions is this inequality sharp”? Band and Boyland tackled this problem

[18], and here is the main result of that paper:

24

Theorem 2.2 (Band and Boyland). For a pA braid β, the Burau estimate is sharp for the

entropy at some root of unity if and only if it is sharp at −1.

2.4 Convention and Notation

The Braid group on (n + 1)-strings {σ1, ...,σn} is traditionally thought of as a

composition of functions, i.e. the braid σ1σ2 is read from right to left so we would perform

σ2 first. However, matrix multiplication is performed from left to the right.

For ease of notation, we would like the order of the Braid group and the Burau

representation to be read in the same way. We are only interested in the spectral radius

of the Burau matrices and a matrix and its transpose have the same spectral radius.

ρ(rb(σ1σ2)) = ρ(rb(σ2) · rb(σ1)) = ρ((rb(σ2) · r(σ1))T ) = ρ(rb(σ1)T · rb(σ2)T )

Notice that the order on the left hand side matches the order on the right hand side. It is

because of this that we will use

r : B(n) → GLn−1(Z[t, t−1])

where r(σi) = rb(σi)T .

25

CHAPTER 3FUNDAMENTAL STIRRING PROTOCOLS

3.1 Definition

For the rest of this paper, we only consider the very special case where we have

n obstacles and a single stirrer. In this specific case, we call stirring protocols the

fundamental stirring protocols. In other words, a fundamental stirring protocol can be

thought of as a pure braid, where only one braid string is allowed to “move” around the

other stationary braid strings.

Definition 5 (Fundamental Stirring Protocol). Consider n distinct points x1, x2, ... , xn

in the interior of D2. We will define a fundamental stirring protocol FSP, by FSP =

(f1(t), ... , fn(t)) where fi : [0, 1] → D2 such that:

1. f1 is continuous,2. fi is constant for each i > 1,3. fi(t) = fj(t) if i = j , for all t ∈ [0, 1],4. fi(0) = xi for each i ,5. f1(1) = f1(0).

The first stirrer associated with f1 will be called a π−stirrer, while the other stirrers

will be referred to as obstacles.

3.2 Configuration Spaces

The fundamental group of the configuration space will give an alternate definition of

the stirring protocol. This new way at looking at a stirring protocol will give insight on the

generators needed to describe the fundamental stirring protocol.

Let M be a manifold of dimension greater than or equal to 2, let∏n

i=1M denote the

n-fold product space, and let F0,n denote the space

F0,n(M) = {(z1, ... , zn) ∈n∏

i=1

M|zi = zj if i = j} (3–1)

Let Qm = {q1, ... qm} be a set of fixed distinguished points of M. Now define Fm,n(M) =

F0,n(M −Qm).

26

Let M = D2 be the disk with the usual Euclidean metric. From Birman, [19] we see

that the fundamental group π1(F0,n(D2)) = Pn where Pn is the pure braid group.

The points {z1, ... , zn} in equation 3–1 are ordered and we can permute them by

acting with the symmetric group,∑

n. The right action µ : F0,n(M) ×∑

n → F0,n is

defined by µ((x0, ... , xn), a) = (xa(0), · · · , xa(n)).∑

n acts freely on F0,n. In general, we

have π1(F0,n(D2))/

∑n = Bn, where

∑n is the symmetric group and Bn is the full braid

group.

Consider the braid group Cn = π1(Fn−1,1), which is a subgroup of Pn (and thus a

subgroup of Bn) describing the case where there are n − 1 obstacles and one stir. We

have the following theorem.

Theorem 3.1. The group formed by the braids generated by a single stirrer with n − 1

obstacles in the disk D2 is isomorphic to Fn−1, the free group on n − 1 letters.

Proof.

Cn(D2) = π1(Fn,1(D

2)) = π1(D2 −Qn−1) = Fn−1

where Fm is the free group generated by m elements.

We can represent braids by a finite number of σj ’s. In our specific case, the first

braid string, (the string to far-left), is allowed to move around the other n − 1 obstacles.

In Figure 3-1, we see an example. Let αk denote the braid, where the first braid string

loops clockwise around the j th string passing in front of the other strings. We can

represent αk in terms of σ’s, as:

αk = σ−11 σ−1

2 σ−13 ...σ−1

k−2σ−1k−1σ

−1k−1σk−2 ...σ2σ1

where, 1 < k ≤ n. Figure 3-1 shows the case where n = 5 and we consider α4.

The fundamental stirring protocol is described by a finite product of αi ’s. If there are

n − 1 obstacles and one fundamental stirrer, then there are 2(n − 1) different αi ’s that

can describe possible paths of the stirrer, namely, {α2,α3, ... ,αn,α−12 , ... ,α−1

n }. There

27

Figure 3-1. The the left stirrer moves around the 4th obstacle

are no relations among the α generators. They form the free group Fn as mentioned in

Theorem 3.1.

Example 1. Consider the case with 4 obstacles and a fundamental stirrer. Let γ be the

stirring protocol where γ = α2α2α3α−14 α5.

Let # denote the number of α’s in the stirring protocol. So for our previous example,

we see that #γ = 5.

We define what is meant by topological efficiency.

Definition 6 (Topological Efficiency). Let γ be a fundamental stirring protocol. Let ϕγ

be the map from the Thurston’s Classification in the isotopy class of the protocol. Let h

denote the topological entropy of ϕγ, then the topological efficiency is

E� (γ) =h(ϕγ)

#γ

A way to think about this definition is the entropy per generator. The entropy will

usually increase with more α’s, however, the efficiency may lower.

28

CHAPTER 4NUMERICAL EXPERIMENTS

To help with the investigation, there were three types of numerical experiments that

were done. The results will be discussed last. From each experiment, we were looking

for patterns and trends with the overall efficiency of the fundamental mixing protocols. In

a paper by Boyland, Aref, and Stremler [20] stated that at least 3 stirrers are needed to

induce a mixing protocol. So we will assume that n is greater than or equal to three.

4.1 Group Actions

We will assume that we have a n−1 obstacles and a fundamental stirrer. Recall that

the group formed by a single stirrer with n − 1 obstacles is π1(D2 − Qn−1) = Fn−1. The

stirrer induces an action on the fundamental group of the disk minus n points, namely,

αk : Fn−1 → Aut(Fn). A direct calculation shows that

αk(xj) =

xj if j < k

(xnxk)−1xj(xnxk) if j = k[

x−1n , x−1

k

]−1xj[x−1n , x−1

k

]if k < j < n − 1

x−1k xjxk j = n

(4–1)

where [a, b] denotes the commutator and 1 ≤ j ≤ n and 1 ≤ k ≤ n − 1. Let L denote the

word length. For example, L (x1x2x−11 x3) = 4. Given a stirring protocol, SP =

∏m

k=1 αnk ,

we can look at the word growth under iteration on an element of Fn−1. One method of

finding the most efficient stirring protocol is to find the stirring protocol that will produce

the largest word in Fn per αi in the protocol. We want to maximize efficiency, not just

entropy.

Let SP =∏m

k=1 αnk be a fundamental stirring protocol. Then we pick an element of

the fundamental group, say xj , and we let SP act on xj by the equations in 4–1. Then we

let SP act on the results and we repeat. Let SPm denote iterating the stirring protocol m

times.

29

Bowen [4] with a theorem in [5] tells us how to compute the entropy of a pseudo-Anosov

map using word growth. Namely,

h(SP) = supxj∈π1(M2)

limm→∞

log(L (SPmxj))/m.

For the experiment, various xj ’s were picked.

If the protocol is truly mixing, i.e. has positive entropy, then the word length will

grow exponentially. A scripting computer language, Perl, was used to test the topological

efficiency of different stirring protocols. However, the program would run out of memory

(2 gigs) after 10 iterations for any protocol that had positive entropy.

4.2 Burau Representation

The second experiment was to compute the Burau representation for the αj ’s

generators with the substitution t = −1 as stated above. Then randomly generated

stirring protocols of different lengths. The entropy was approximated by the logarithm of

the spectral radius. This program was written in Mathematica.

4.3 Mousifir Algorithm

The previous two methods only gave approximations. The next one gives the exact

entropy. The two programs were written by Toby Hall [21].

The first program trains.exe computes the entropy of a braid by using the Bestvina-Handel

algorithm. The second program dynn.exe computes the entropy of a braid by using the

Mousifir Algorithm.

Random fundamental stirring protocols were created and were turned into

braids as described above by a C++ program and then inputed that word into Toby

Hall’s programs. The output was analyzed by Excel where topological efficiency was

computed.

4.4 Results of Experimentation

Every experiment yielded the same result, namely the protocol αn−1αn−2 · · ·α2 has

the largest efficiency for n − 1 obstacles with a single stirrer. We will call this protocol

30

HSPn. Remember that the fundamental stirrer can also be considered as a braid string,

so HSPn has a total of n braid strings or in other words, the n − 1 obstacles and the

fundamental stirrer.

If we change the positions of the open holes, we see that HSP has a lot symmetry.

In Figure 4-1 we see the case for HSP6.

Figure 4-1. Two versions of HSP6

Another interesting fact that came out, is that the efficiency does not depend on

essential crossings, the minimal number of times the fundamental stirrer crosses its own

path, of the stirring protocol.

Let us compare the two following stirring protocols in Figure 4-2. The stirring

protocol on the left is HSP6, while the stirring protocol on the right uses the same alpha’s

as the left but in a different order. The protocol on the left has three times the amount of

topological entropy as the one on the right.

Figure 4-2. The left stirring protocol is more efficient.

In a paper by Finn, Cox and Bryne [8], a numerical experiment was run on random

stirring protocols, without obstacles. Specifically, HSP8’s path was studied but minus any

31

obstacles. Their experiment focused on specific fluids whereas the HSP was derived

without specific fluid properties in mind. The authors made a note in their paper, that

HSP8 protocol did not traverse through the center of the fluid, while their other protocols

did, and yet HSP8 still performed remarkably well. They also used different measures to

determine mixing, like diffusion, which is not considered here.

32

CHAPTER 5PROPERTIES OF HSP

From the experiments we conjecture the following:

Conjecture 1. The most efficient fundamental stirring protocol with n − 1 obstacles is

HSPn = α2 ...αn.

5.1 Entropy Lower Bound for HSP

In this section we give some of the properties of HSPn. In Figure 5-1 we see the

behavior of the efficiency of HSP over n where n is the number of obstacles.

Figure 5-1. Efficiency of HSP

We will denote HSPn as the Burau representation of HSPn evaluated at t = −1.

Theorem 5.1. Let HSPn be the stirring protocol defined in the previous chapter. Then

the trace of HSPn is −3n−1 + 3n − 2.

Proof. First we will introduce notation. Recall the σi notation from the Artin Braid in

section 1.2. We will now keep track of the total number of strings. So for σi in a braid

group with a total of n-strings, we will denote as σi ,n. Also, for simplicity, we will use a hat

over a braid to denote the reduced Burau representation evaluated at t = −1.

Consider the reduced Burau representation for the braid generators for n braids

{σ1,n, σ2,n, ... , σn−1,n} and compare that to the generators for n + 1 braids

{σ1,n+1, σ2,n+1, ... , σn−1,n+1, σn,n+1}.

33

By the definition of the reduced Burau representation and 1 ≤ i ≤ n − 1 we have,

σi ,n+1 =

0

σi ,n...

0

0 ... 0 1

Now, we will also define αi ,n in a similar manner as the σi ,n, where the additional k tells

us the total number of strings. For 2 ≤ k ≤ n we have the following:

αk,n = σ−11,n · σ−1

2,n · σ−13,n ...σ

−1k−2,n · σ

−1k−1,nσ

−1k−1,n · σk−2,n ...σ2,n · σ1,n

For 2 ≤ j ≤ n − 1 we have

αj ,n+1 =

0

αj ,n

...

0

0 ... 0 1

34

Now, we just have to find αn,n+1 and αn+1,n+1. These cases are different because the last

row of αn,n+1 and αn+1,n+1 have more nonzero entries. First we will compute, αn,n+1.

αn,n+1

= (σ1,n+1 · σ2,n+1 ... σn−2,n+1)σ−2n−1,n+1(σ1,n+1 · σ2,n+1 ... σn−2,n+1)

−1

= (σ1,n+1 · σ2,n+1 ... σn−2,n+1)

0

In−1

...

0

0 ... 0 2 1 −2

0 ... 0 1

(σ1,n+1 · σ2,n+1 ... σn−2,n+1)

−1

=

0 0 0 ... 0 1 1

1 0 0 ... 0 1 1

0 1 0 ... 0 1 1

0 0 1 ... 0 1 1

...

0 0 ... 0 1 1 1

0 0 0 ... 0 0 1

1 0 0 ... 0 0 0

0 1 0 ... 0 0 0

0 0 1 ... 0 0 0

...

0 0 0 ... 1 0 0

0 0 ... 0 2 1 −2

0 0 0 ... 0 0 1

0 0 0 ... 0 1 1

1 0 0 ... 0 1 1

0 1 0 ... 0 1 1

0 0 1 ... 0 1 1

...

0 0 ... 0 1 1 1

0 0 0 ... 0 0 1

−1

=

−2

αn,n

...

−2

0 ... 0 1

35

Since the HSPn = αn,n · αn−1,n · αn−2,n · · · α2,n, we have that

HSPn+1 = αn+1,n+1 ·

−2

HSPn

...

−2

0 ... 0 1

where

αn+1,n+1 =

3 0 · · · 0 −2

2 −2

... In−2

...

2 −2

2 0 · · · 0 −1

and In−2 denotes the n − 2 by n − 2 identity matrix.

Then by construction,

HSPn+1 = αn+1 ·

−2

HSPn

...

−2

0 ... 0 1

HSPn(i , j) =

3HSPn−1(1, j) i = 1 and 1 ≤ j ≤ n − 2

−8 i = n − 1 and 1 ≤ j ≤ n − 2

2HSPn−1(1, j) + HSPn−1(i , j) 2 ≤ i ≤ n − 2 and 2 ≤ j ≤ n − 2

2HSPn−1(1, j) 1 ≤ i ≤ n − 2 and j = n − 1

−5 i = n − 1 and j = n − 1

From this complicated description, we can see a formula for the trace Tr(HSPn).

For convenience, let An = HSPn and let Tn = Tr(An).

36

Tn = An(1, 1) + · · ·An(n, n)

= 3An−1(1, 1) + 2

n−2∑k=2

An−1(1, k) +

n−2∑k=2

An−1(k , k)− 5

= 2An−1(1, 1) + 2

n−2∑k=2

An−1(1, k) +

n−1∑k=1

An−1(k , k)

= 2An−1(1, 1) + 2

n−2∑k=2

An−1(1, k) + Tn−1

Thus, we have the following equation for the trace,

Tn = 2An−1(1, 1) + 2

n−2∑k=2

An−1(1, k) + Tn−1 (5–1)

Let Dn−1 =∑n−2

k=2 An−1(1, k). Note that the subscript is used to indicate the sum of the

top row for An−1. By the inductive construction, we see that

D3 = −8 (5–2)

D4 = −8− 3 · 8 (5–3)

D5 = −8− 3 · 8− 32 · 8 (5–4)

... (5–5)

Dn−1 = −8

n−4∑k=0

3k =−4(3(n−1) − 9)

9(5–6)

Also, the first term of 5–1 has a pattern. Again, we start with n = 3.

A3(1, 1) = 3 (5–7)

A4(1, 1) = 32 (5–8)

... (5–9)

An−1(1, 1) = 3n−3 (5–10)

37

When we substitute, 5–10 and 5–6 back into 5–1 we have the following equation:

Tn = −2 · 3n−2 + 3 + Tn−1 (5–11)

When n = 3, we know that T3 = −2, our initial condition. Building from our initial

condition and 5–11, we have,

Tn = −3n−1 + 3n − 2 (5–12)

From this theorem we have a corollary about the efficiency of HSPn. This corollary

uses the concept of train tracks in order to invoke a theorem by Band and Boyland [18].

A good source for train tracks include [15], [22] and [23]. Suppose f : M2 → M2, then

the idea of a train track for f is an embedded graph on the surface M2 that satisfies

several conditions and is invariant under f .

Corollary 1. Let HSP be the stirring protocol defined as above. Then

lim supn→∞

E� (HSPn) ≥ ln(3)

Proof. By a direct calculation the train track of each HSPn has a singularity structure

which consists of just one-prongs at the punctures and the usual singularity at infinity.

By the paper by Band and Boyland [18] (theorem 5.1 in the paper), we see that this

implies the Burau estimate for HSPn is sharp for the substitution t = −1.

Figure 5-2. Train Track for the case n = 5. The red disk denotes the stirrer.

As before, let HSPn be the Burau representation evaluated at t = −1. Let ρ(HSPn)

denote the spectral radius of HSPn. Let λ1, · · · ,λn−1 denote the eigenvalues of HSPn.

38

Note that there are only n − 1 eigenvalues since we are using the reduced Burau

representation. Without loss of generality, assume that ρ(HSPn) = |λ1|.

|Tr(HSPn)| = |λ1 + λ2 + · · ·λn−1|

≤ |λ1|+ |λ2|+ · · · |λn−1|

≤ (n − 1)|λ1|

Thus by the previous theorem 5.1 we have,

ρ(HSPn) = |λ1| ≥|Tr(HSPn)|

n − 1=

∣∣∣∣−3n−1 + 3n − 2

n − 1

∣∣∣∣ = 3n−1 − 3n + 2

n − 1

Now we have a bound for the efficiency.

lim supn→∞

E� (HSPn) = lim supn→∞

log(ρ(HSPn))

n − 1

≥ limn→∞

log

(3n−1 − 3n + 2

n − 1

)n − 1

= limn→∞

log(3n−1 − 3n + 2)− log(n − 1)

n − 1

= log(3)

To obtain equality in the corollary, we need to know more about the eigenvalues.

For example, if knew that the sum over all the eigenvalues, except the one that is

the spectral radius, grew linearly, then we would know that the asymptotic limit of

the efficiency of HSPn is exactly log(3). It turns out that the first few characteristic

polynomials of HSPn are palindromic/anti-palindromic Salem polynomials. A good

reference for Salem polynomials is [24]

Definition 7 (palindromic/anti-palindromic polynomials). A polynomial p(x) ∈ Z[x ] is said

to palindromic if it satisfies

p(x) = xmp(1/x)

39

where m is the degree of p(x). If the polynomial satisfies

p(x) = −xmp(1/x)

where m is the degree of p(x) then p(x) is said to be anti-palindromic.

In other words, if the polynomial is in standard form and the coefficients are the

same when read frontwards as well as backwards we say the polynomial is palindromic.

Anti-palindromic is the same as palindromic, except that the coefficients differ by a

negative sign.

Definition 8 (Salem number). A Salem number is a real algebraic integer, greater than

1, with the property that all its Galois conjugates lie on or within the unit circle, and at

least one Galois conjugate lies on the unit circle. An irreducible polynomial that has a

root that is a Salem number is called a Salem Polynomial.

Let fn(x) be the characteristic polynomial for HSPn. Here are just the first few

polynomials:

f3(x) = 1 + 17x − 17x2 − x3 = −(x − 1)(1 + 18x + x2)

f4(x) = 1 + 68x − 122x2 + 68x3 + x4

f5(x) = 1 + 227x − 542x2 + 542x3 − 227x4 − x5

= −(x − 1)(1 + 228− 314x2 + 228x3 + x4)

It appears that the characteristic polynomials are either palindromic or anti-palindromic

based on their parity. It turns out that any Burau matrix with t = −1 enjoys this property.

Let Bn(t) be a n by n matrix and Burau representation. Let �Bn(t) = Bn(1/t), and

B∗n(t) = �Bn(t)

T where T is the transpose. Thus, �Bn(−1) = Bn(−1) We may assume that

B is the reduced Burau representation (n − 1) by (n − 1).

40

Theorem 5.2 (Squire). The Burau Representation is Unitary, i.e. There exists a non-

singular matrix J ∈ GL(n − 1,Z) such that

B∗JB = J

The proof may be found in the paper by Squier [19]

Theorem 5.3. The characteristic polynomial of B when t = −1 is either palindromic or

anti-palindromic depending on the parity of n − 1.

Proof.

char(B)(x) = det(B − xI)

= det(J−1(B∗)−1J − xI) = det((B∗)−1 − xI)

= det((B∗)−1 − B∗xI(B∗)−1)

= det(I− xB∗) det((B∗)−1)

= det(B∗)−1 det(I− xB∗)

= det(B∗)−1 det(−x(B∗ − x−1I))

=(−x)n−1

det(B∗)det(B∗ − x−1I)

So det(B∗) = det(�BT ) = det(�B) = det(�σk1�σk2 ... �σkm) = det(�σk1) · · · det(�σkm) When

t = −1, then det(σki ) = 1, so det(B∗) = 1. Let B−1 denote B with t = −1.

Thus,

41

char(B−1)(x) = (−x)n−1 det(B∗−1 − x−1I)

= (−x)n−1 det(�BT−1 − x−1I)

= (−x)n−1 det

((�B−1 − x−1I)T

)= (−x)n−1 det(B−1 − x−1I)

= (−x)n−1 det(B−1 − x−1I)

= (−x)n−1char(B−1)(1/x)

Let p(x) = char(B)(x), then we have two cases.

1.If n − 1 is even, then p(x) = xn−1p(1/x) and p is palindromic.2.If n − 1 is odd, then p(x) = −xn−1p(1/x) and p is anti-palindromic.

We should note that if p(x) is an anti-palindromic polynomial, then p(1) = −p(1)

implies that p(1) = 0. Thus, if n − 1 is odd, then char(B)(x) = (x − 1)q(x) for some

polynomial q(x).

Theorem 5.4. Suppose B is a Burau Matrix with t = −1 that is (n − 1) × (n − 1) where

n − 1 is odd. Then char(B)(x) = (x − 1)q(x) where q(x) is palindromic.

Proof. First, we see that char(B)(x) = (x −1)q(x) and that the degree of q is n−2, thus,

char(B)(x) = −xn−1char(B)(1/x)

(x − 1)q(x) = −xn−1(1/x − 1)q(1/x)

x(x − 1)q(x) = −xn−1(1− x)q(1/x)

x(x − 1)q(x) = xn−1(x − 1)q(1/x)

(x − 1)q(x) = xn−2(x − 1)q(1/x)

q(x) = xn−2q(1/x)

42

Thus, every stirring protocol will have a Burau representation with t = −1 whose

characteristic polynomials are palindromic/anti-palindromic polynomials. However, the

first few characteristic polynomials for HSP are nicer still. They are Salem polynomials.

Using the technique from [24], we see that a direct computation shows that the

first few characteristic polynomials of the Burau representation with t = −1 for HSP are

Salem polynomials.

Conjecture 2. The characteristic polynomial of the Burau representation with t = −1 of

HSPn is a Salem polynomial.

If HSPn has a characteristic polynomial that is a Salem polynomial then we would

have sharpness in corollary 1. Since HSP is constructed inductively, as seen above, it

would be nice if the characteristic polynomials also enjoyed that property.

Conjecture 3. Let gn(x) be the characteristic polynomial for HSP(n), then

gn+1(x) = gn(x)q(x) + gn−1(x)p(x)

where q(x) = −4(x − 1) and p(x) = −(3x2 − 10x + 3).

This conjecture was based on the assumption that the characteristic polynomials

were inductive and has been verified for the first 20 polynomials.

5.2 Gearability of HSP

One concern about fluid mixing and stirring protocols is the practicality of the stirring

device itself. For example, a stirring protocol would be useless if the machine used

to stir the fluid was too complicated. To quote Kobayashi and Umeda in [25] “...from

the viewpoint of practical mixing, realizing such a device via simple mechanism, or

mechanism requiring less energy is an important issue.”

Definition 9 (Gearable). A stirring protocol is called gearable if a device using a finite

set of gears can move a stirring rod along the path of said stirring protocol.

43

We will now show that HSPn is in fact gearable by showing that the path of the stirrer

follows that of a hypotrochoid.

Suppose we have two circles c1 and c2 where c2 is centered at the origin and the

radius of c1 is r and c2 is R where R > r . We suppose that c1 goes along the inside of

c2 in a counter clockwise direction without slipping. A point p is the d distance from the

center of c1. As c1 travels around c2, the path of p is the path of the hypotrochoid. Below

are the parametric equations where θ goes from 0 to 2π.

x(θ) = (R − r) cos(θ) + d cos

(R − r

rθ

)y(θ) = (R − r) sin(θ)− d sin

(R − r

rθ

)In the case of HSPn+1, we have R = 1, r = 1/n and d = 2/n. The figure 5-3 is an

example of HSP8 as a hypotrochoid. Now we can explain the encrypted letters in the

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

Figure 5-3. HSP(8): The 7 obstacles are the blue dots and the path of the fundamentalstirrer is red.

HSP which stand for Hypotrochoid Stirring Protocol. In fact, Kobayashi and Umeda

in [25] actually studied different types of hypotrochoid stirring protocols specifically

because of the practicality.

44

CHAPTER 6UPPER BOUNDS

6.1 Construction of Upper Bound

Given a fundamental stirring protocol with n − 1 obstacles and a stirrer, we will now

give an explicit upper bound for the efficiency which we defined above as

E(n) = sup{E� (β) : β is a fundamental stirring protocol with n − 1 fixed holes }.

Even though there are n − 1 fixed holes or obstacles, the stirrer also counts an obstacle.

So the n in E(n) is counting all the obstacles in the fluid.

We computed a lower bound in the previous chapter, namely,

log((3n−1 − 3n + 2)/(n − 1))

n − 1≤ log(ρ(HSPn))

n − 1≤ E(n).

The construction of the upper bound uses the definition of the generalized spectral

radius. Let M be a finite set of matrices and ρ denote the spectral radius of a matrix, i.e.

the magnitude of the largest eigenvalue. Define

ρk(M ) = maxM1,...,Mk∈M

ρ(M1 · · ·Mk)1/k

In other words, for a fixed k value, we consider all products that use k matrices from the

set M . Since M is a finite set, then the set of all products of length k is also finite. From

this new set of matrices, take the spectral radius of each matrix and find the maximum.

That maximum value is ρk(M ).

Definition 10 (Generalized Spectral Radius).

ρ(M ) = lim supk→∞

ρk(M )

This definition first appears in [26]. For properties of matrices that are used in this

section, please refer to the appendix A.1.

45

We will also need the following property of logarithms. This property is commonly

used in dealing with topological entropy and a good reference is [27].

Lemma 1. Suppose {an} and {bn} are two sequences of positive real numbers such

that limn→∞(log an)/n = a and limn→∞(log bn) = b exist. Then

limn→∞

log(an + bn)/n = max(a, b).

Proof. Since an and bn are positive real numbers we have that log(an + bn)/n ≥ log(an)/n

and log(an + bn)/n ≥ log(bn)/n for each n. Thus limn→∞ log(an + bn)/n ≥ max(a, b).

To show the reverse inequality, fix c > max(a, b). Thus, c > log(an)/n and

c > log(an)/n for each n implies that an < ecn and bn < ecn for each n. Now we have

log(an + bn)

n<

log(2ecn)

n= c +

log(2)

n

Taking limits, we see that limn→∞ log(an + bn)/n ≤ c . This inequality holds for any

c > max(a, b). We must have limn→∞ log(an + bn)/n ≤ max(a, b).

Theorem 6.1. E(n) ≤ log(3)

Proof. To obtain this upper bound, it is convenient to replace the generators of the

fundamental group of the (n+1) punctured disk. The new generators δi in the case n = 6

are shown in the figure 6-1. In figure 6-2 we show α4 acting on the new generators.

Recall that α4 sends the fundamental stirrer around the 4th obstacle. In general, we

have two cases:

(αj)∗(δj) = (δj−1δj−2 ... δ2δ−11 δn ... δj+1)

−1δj(δj−1δj−2 ... δ2δ−11 δn ... δj+1) (6–1)

(αj)∗(δk) = δk , when k = j (6–2)

46

Figure 6-1. The new generators for π1(M2) with n = 6; the square is the fundamentalstirrer and the circles are the obstacles

Figure 6-2. The α4 acting on the new generating set with n = 6

47

If β∗ : π1(M2) → π1(M

2) is the induced action of the pseudo-Anosov braid β on π1(M2),

then by Bowen [4] and Fathi [5] we have

h(β) = supg∈π1(M2)

limk→∞

log(L (βk∗g))

k

where L (βk∗g) denotes the number of generators in the reduced word of βk

∗g.

Define ni(g) : π1(M2) → N where ni(g) is the number of occurrences of the

generators δi and δ−1i in the reduced word for g. Let n(g) = (n1(g), ... , nn(g)). Thus,

L (g) = ∥n(g)∥1.

We will use a simpler form of h(β) from the formula in 6.1 namely,

µ(β) = maxδ1,...,δn

limk→∞

log(∥n(βk∗ δi)∥1)

k.

We will now show that µ(β) = h(β), i.e. we can take the maximum over the generators

instead of the supremum over all words.

It is obvious that µ(β) ≤ h(β) since δ1, ... , δn ∈ π1(M2). Now we will use lemma

6.1 to show the reverse inequality. Let g ∈ π1(M2) be an arbitrary element where

g = δϵi1i1

... δϵijij

where ϵn ∈ {±1}. We also need to point out that L (βk∗ (δi)) = L (βk

∗ (δ−1i ))

for each i because the β∗(δi) has the same number of occurrences of the generators as

β∗(δ−1i ) for each i . Thus, ∥n(βk

∗ (δi))∥1 = ∥n(βk∗ (δ

−1i ))∥1 for each i .

limk→∞

log(L (βk∗ (g)))

k= lim

k→∞

log(L (βk∗ (δ

ϵi1i1

... δϵijij)))

k

≤ limk→∞

log(L (βk∗ (δ

ϵi1i1) + · · ·+ L (βk

∗ (δϵijij))))

k

= limk→∞

log(L (βk∗ (δi1) + · · ·+ L (βk

∗ (δij ))))

k

≤ maxδi1 ,...,δij

limk→∞

logL (βk∗ (δi))

k(By lemma 6.1.)

≤ maxδ1,...,δn

limk→∞

log(∥n(βk∗ δi)∥1)

k(Maximum over all generators.)

= µ(β)

48

Since β was arbitrary, we see that h(β) ≤ µ(β).

So we have that h(β) = µ(β), i.e.

h(β) = maxδ1,...,δn

limk→∞

log(∥n(βk∗ δi)∥1)

k.

Definition 11 (Incidence Matrix). Using the specified set of generators for π1 as in 6-1

and for any homomorphism α∗ : π1(M2) → π1(M

2) define the incidence matrix A by

Ai ,j = nj(α∗δi).

Let A be the incidence matrix for α∗. Notice that the j th component of n(g)A is the

number of occurrences of δj and δ−1j before any cancellations are done. Thus,

n(g)A ≥ n(α∗g)

where [a1, ... , am] ≥ [b1, ... , bn] means ai ≥ bi for all i . When we iterate, we get for all

k > 0,

n(g)Ak ≥ n(αk∗g).

Taking one-norms (everything is positive) we have

∥n(αk∗g)∥1 ≤ ∥n(g)Ak∥1 ≤ ∥n(g)∥1∥Ak∥1.

Hence,

limk→∞

∥n(αkg)∥1/k1 ≤ limk→∞

∥n(g)∥1/k1 ∥Ak∥1/k1 = ρ(A)

where ρ is the spectral radius of the matrix A.

Thus, for any β from a fundamental stirring protocol, if its incidence matrix is Aβ then

h(β) ≤ log(ρ(Aβ)).

Recall that the fundamental stirrer is counted as the first hole, and αi denotes braid

the element sending the first string around the i th string and i ∈ {2, ... , n}. Using (6–1)

49

and (6–2) we have that corresponding incidence matrix is:

Aαi=

1 0 ... 0 0 0 ... 0

0 1 0 ... 0 0 ... 0

...

2 2 ... 2 1 2 ... 2

0 0 ... 0 0 1 ... 0

...

0 0 ... 0 0 0 ... 1

(i − 1)th row

Also note that αi and α−1i have the same incidence matrices since nj(αiδi) = nj(α

−1i δi)

for every i and j .

For ease of notation, let Ai = Aαi−1 so A1 = Aα2, for example. Let M =

{A1, ... ,An−1}. Let β be a fundamental stirring protocol with word length m so β =

αk1 ...αkm and let Aβ denote the incidence matrix for β. Then for each i and j we have,

(Aβ)ij = nj(βδi)

= nj(αk1 ...αkm(δi))

≤ nj(αk1δi) · · · nj(αkmδi) Since n is counting before cancellations

= (Ak1)ij · · · (Akm)ij .

In other words, Aβ ≤ Ak1 · · ·Akm where ≤ means for each ij-th position. Then

ρ(Aβ) ≤ ∥Aβ∥1 ≤ ∥Aαi1∥1 ... ∥Aαim

∥1 ≤ 3m

Since β was an arbitrary fundamental stirring protocol with length m then,

ρm(M ) = maxA1,...,Am∈M

ρ(Am · · ·A1)1/m ≤ (3m)1/m = 3

and

ρ(M ) = lim supk→∞

ρk(M ) ≤ 3.

50

We now have an upper bound for the topological efficiency (recall the definition of

topological efficiency from section 3.2) of any arbitrary stirring protocol β, namely,

E� (β) =h(β)

#β≤ log(ρ(Aβ))

#β.

Since #β is number of alpha generators used to construct β, then #β is also the

number of incidence matrices used to construct Aβ. Thus,

E� (β) ≤ log(ρ(Aβ)1#β ) ≤ log(ρ(M )) ≤ log(3)

This is true for every fundamental stirring protocol and by taking the sup over every

fundamental stirring protocol we have, E(n) ≤ log(3).

6.2 Spectral Radius of incidence Matrices

Let HSPn = Aαn...Aα2. So HSPn is the product of the incidence matrices of the α

stirrers in the same order as the HSP(n). We will explore some properties of HSPn.

HSPn =

1 2 ... 2

0 1 ... 0

0 0 1 ... 0

... ......

0 0 0 ... 1

1 0 ... 0

0 1 0 ... 0

0 0 1 ... 0

... ......

0 0 0 ... 1

· · ·

1 0 ... 0

2 1 2 ... 2

0 0 1 ... 0

... ......

2 2 2 ... 1

HSPn =

2 · 3n−1 − 1 2 · 3n−1 − 22 · 30 2 · 3n−1 − 22 · 31 ... 2 · 3n−1 − 22 · 3n−2 2 · 3n−2

2 · 3n−2 ... 2 · 3n−3

2 · 3n−3 ... 2 · 3n−4

... ......

2 · 3 ... 2 · 30

2 2 2 2 ... 1

We will let ρ(HSPn) denote the spectral radius of HSPn and we let {λ1,λ2, ... ,λn}

denote the eigenvalues of HSPn and λ1 is the Perron-Frobenius value, i.e. ρ(HSPn) = λ1

51

First, we will compute the column sums. Let Ci denote the column sum for the i th

column. We notice that

C1 = 2 · 3n−1 − 1 + 2 · 3n−2 + · · ·+ 2 · 3 + 2 = −1 + 2

n−1∑j=0

3j = 3n − 2 (6–3)

Cn = 2 · 3n−2 + 2 · 3n−3 + · · ·+ 2 · 30 + 1 = 1 + 2

n−2∑j=0

3j = 3n−1 (6–4)

The thing to note is that column sums are decreasing. So C1 ≥ C2 ≥ · · ·Cn. Next,

we consider that ρ(HSPn) = ρ(HSPn

T) where HSPn

Tdenotes the transpose of HSPn.

From [28] and definition 17 from the appendix and , we have

Theorem 6.2. Suppose that M is an n × n non-negative primitive matrix. Then,

mini

n∑j=1

mij ≤ ρ(M) ≤ maxi

n∑j=1

mij

with equality on either side implying equality throughout. A similar proposition holds for

column sums.

So in our case, this implies

minj

n∑i=1

(HSPn

T)i ,j ≤ ρ(HSPn

T) ≤ max

j

n∑i=1

(HSPn

T)i ,j

which implies that

Cn ≤ ρ(HSPn) ≤ C1

3n−2 ≤ ρ(HSPn) ≤ 3n − 2.

Definition 12 (Pisot number). A Pisot number is a real algebraic integer, greater than

1, with the property that all its Galois conjugates lie within the unit circle. An irreducible

polynomial that has a root that is a Pisot number is called a Pisot Polynomial.

Conjecture 4. The matrix HSPn has a characteristic polynomial whose roots are Pisot.

We see that (HSPn)j ,j ≥ 0 for each j . Thus, we have

|2 · 3n−1| = |(HSPn)1,1| ≤ |Tr(HSPn)| = |λ1 + λ2 + · · ·+ λn| ≤ |λ1|+ |λ2|+ · · ·+ |λn|

52

Supposing that the conjecture is true, then |λi | ≤ 1 for each i ≥ 2. So we have

2 · 3n−1 ≤ |λ1|+ (n − 1)

Thus, 2 · 3n−1 − n − 1 ≤ ρ(HSPn).

Let M be the set of the incidence matrices for the fundamental stirring protocols αi

where i ∈ {2, ... , n}, i.e.

M = {Aα2, ... ,Aαm}.

For ease of notation, we will use Aαi= Ai−1 and An = Aαm

, so M = {A1, ... ,An}. We

have already seen that ρ(M ) ≤ 3.

We would like to compute ρ(M ) directly. In order to prove this we need to setup

some machinery. The main idea for computing ρ(M ) is that for any positive matrix N, Ai

acts very simply on the vector of columns sums of N for each i .

We use the joint spectral radius which we will define now. Recall the definition of an

induced matrix norm from definition 19 in the appendix. Let M be a finite set of matrices

and ∥ · ∥ denote an induced matrix norm. Define

�ρk(M , ∥ · ∥) = maxA1,...,Ak∈M

∥M1 · · ·Mk∥1/k

Definition 13 (Joint Spectral Radius).

�ρ(M , ∥ · ∥) = lim supk→∞

�ρk(M )

The Joint Spectral Radius is independent of the norm that we choose [29]. So for

simplicity we will use �ρ(M ) for the Joint Spectral Radius for the set M . Also, if M is a

finite set of matrices, then ρ(M ) = �ρ(M ) which was first proved in [30].

Define c : GLn(Z) → Zn where c(M) is the column sums of M.

Example 2. By direct calculation we have, c(Ai) = (3, 3, ... , 3, 1, 3, ... , 3) where the 1 is

in the i th place.

Lemma 2. Let M ∈ GLn(Z) be an arbitrary matrix, and Ai ∈ M , then c(MAi) = c(M)Ai .

53

Proof. Fix i ∈ {1, ... , n} and consider MAi .

MAi =

m11 m12 ... m1n

......

mi1 min

......

mn1 mn2 mnn

1 0 ... 0

......

2 2 · · · 2 1 2 · · · 2...

...

0 0 1

=

m11 + 2m1i ... m1i ...m1n + 2m1i

......

mi1 + 2mii mii min + 2mii

......

mn1 + 2mni ... mni ...mnn + 2mni

By computing the column sums, we have

c(MAi) = (c1(M) + 2ci(M), ... , ci−1(M) + 2ci(M), ci , ci+1(M) + 2ci(M), ...

... , cn(M) + 2ci(M)). (6–5)

Next, we will compute the c(M)Ai .

c(M)Ai = (c1(M), ... cn(M))

1 0 ... 0

......

2 2 · · · 2 1 2 · · · 2...

...

0 0 1

= (c1(M) + 2ci(M), ... , ci−1(M) + 2ci(M), ci , cj+1(M) + 2ci(M), ...

... , cn(M) + 2ci(M)). (6–6)

Thus, (6–5) and (6–6) are the same so c(MAi) = c(M)Ai and since i was arbitrary, our

lemma is true.

54

Fix n ≥ 3 and let On be all lists of n positive integers given in decreasing order, so

On = {a ∈ (Z+)n : a1 ≥ a2 ≥ ... an > 0}

Definition 14 (Sorting Function). Let S : (Z+)n → On be the sorting function, i.e. it sends

a vector of positive integers to the list of its entries sorted in non increasing order.

Example 3. Let a = (1, 2, 3) then S (a) = (3, 2, 1).

Let a ∈ (Z+)n, then Ai · aT ⊂ (Z+)n. Now, we define

Wj : On → On where Wj (a) := S ◦ Aj (aT ).

We will also need the standard order on On, namely,

a ≤ b ⇐⇒ aj ≤ bj for all j

a < b ⇐⇒ aj < bj for all j

Next, we will have a lemma that will go with this ordering.

Lemma 3. Assume that a = (a1, a2, ... , am) and b = (b1, b2, ... , bm) where a > 0 and

b > 0. Then the following are true.

(a) a ≥ b ⇒ Wj (a) ≤ Wj(b) for all j(b) a > b ⇒ Wj (a) > Wj(b) for all j(c) j < k and aj = ak ⇒ Wj (a) = Wk(b)

(d) j < k and aj > ak ⇒ Wj (a) > Wk(b)

(e) j ≤ k and a > b ⇒ Wj (a) > Wk(b)

Proof. Assume that a = (a1, a2, ... , am) and b = (b1, b2, ... , bm) and ai > 0 and bi > 0.

55

For part a) suppose a ≥ b, then ai ≥ bi for each i .

Wj (a) = S ◦ Aj (aT ) = S(a1 + 2aj , ... , aj−1 + 2aj , aj , aj+1 + 2aj , ... , am + 2aj)

= (a1 + 2aj , ... , aj−1 + 2aj , aj+1 + 2aj , am + 2aj , ... , aj)

Wj(b) = S ◦ Aj(bT ) = S(b1 + 2bj , ... , bj−1 + 2bj , aj , bj+1 + 2bj , ... , bm + 2bj)

= (b1 + 2bj , ... , bj−1 + 2bj , bj+1 + 2bj , bm + 2bj , ... , bj)

Since ai ≥ bi for each i , then ai + 2aj ≥ bi + 2bj for each i . Thus, Wj (a) ≥ Wj(b).

Part b) is similar to part a).

For part c) suppose we have a where j < k and aj = ak . Then,

Wj (a) = S ◦ Aj (aT ) = S(a1 + 2aj , ... , aj−1 + 2aj , aj , aj+1 + 2aj , ... , ak + 2aj , ... , am + 2aj)

= (a1 + 2aj , ... , aj−1 + 2aj , aj+1 + 2aj , am + 2aj , ... , ak + 2aj , ... , aj)

Wk (a) = S ◦ Ak (aT ) = S(a1 + 2ak , ... , aj−1 + 2ak , ak , aj+1 + 2bk , ... , am + 2ak)

= (a1 + 2ak , ... , aj + 2bk , ... ak−1 + 2bk , ak+1 + 2bk , am + 2ak , ... , bk)

Since aj = ak , then ai + 2aj = ai + 2ak for all i . Thus, Wj (a) ≥ Wk (a).

Part e) is similar to part c).

Definition 15 (Length-k Strategy). A length k strategy is a list �s = (j1, j2, ... , jk) with

1 ≤ ji ≤ n for all i where each �s defines a map W�s := Wjk ◦ · · · ◦Wj1 so W�s : On → On.

Definition 16 (Standard with respect a). Fix an a ∈ On and say �s is standard with

respect to a if for each i we have

b = Wji−1 ◦ · · · ◦Wj1 (a)

where either ji = 1 or bji−1 > bji .

Proposition 6.1. Given a ∈ On, if �s is a length-k strategy and is standard for a and

�s = �1, then

W�1(a) > W�s (a).

56

Proof. Let �s = (j1, ... , jk) and let i be the first index where ji > 1. Let

b = Wji−1,...,j1.

Since s is standard for a, then bji < bji−1 ≤ b1. Thus, by lemma 3 d, W1(b) > Wji (b), i.e.

W1,...,1(a) > Wji ,...,j1(b).

Since jn ≥ 1 for all n > i , then by lemma 3 e, then W1,...,1(a) > W�s , as required.

Corollary 2. If �s is any length k strategy for a then W�1 ≥ W�s .

Proof. If �1 is the standard strategy derived from �s, then W�1(a) = W�s (a) by lemma 3 a.

Otherwise use the previous proposition.

Theorem 6.3. Let M = {A1, ... ,An} be as defined above. Then the joint spectral radius

�ρ(M ) = �ρ(HSPn), i.e. the spectral radius of M is realized by HSPn.

Proof. Fix w > 0. We consider an arbitrary number w of matrices from the set M . Let

N = Ai1 · · ·Aiw . By the division algorithm, we have w = m _n + j where 0 ≤ j < n. The

theorem follows from two observations.

First, for the matrix N, there is a standard strategy �s for a = (1, 1, ... , 1), so that

W�s (a) = c (aN) = c (aAi1, ... ,Aiw ) = ∥N∥1.

Second,

c (a A1, ...An,A1, ... ,An, ... ,A1, ... ,Aj︸ ︷︷ ︸m _n+j elements

) = W�1(a)

of length mn + j .

Then, by corollary 2, we have

c (aA1, ...An,A1, ... ,An, ... ,A1, ... ,Aj) ≥ c (aAi1, ... ,Aiw )

57

where w = mn + j . In particular, since for a positive matrix, c(A) = ∥A∥1 and c (aA) =

c(IA) = c(A), we have that

∥A1 · · ·AnA1 · · ·An · · ·A1 · · ·Aj∥ ≥ ∥Ai1Aiw∥1.

Since, Ai1 · · ·Aiw is constructed from w = mn + j elements and was arbitrary, then by

definition,

�ρw(M ) = maxAj1

,...,Ajw

∥Aj1, ... ,Ajw∥1 = ∥A1, ...An,A1, ... ,An, ... ,A1, ... ,Aj∥1/w1 .

Thus,

�ρ(M ) = limk→∞

�ρk(M ) = limk→∞

∥(A1, ...An)k∥1/k1 = ρ(A1, ... ,An).

By Berger and Wang in [30], it was shown that the generalized spectral radius and

the joint spectral radius are equal for bounded sets of matrices. In our case, we have

that,

ρ(M ) = �ρ(M ).

Corollary 3.log(ρ(HSPn))

n − 1≤ E(n) ≤ log(ρ(HSPn)).

58

APPENDIX: SOME GENERAL MATRIX LEMMAS

A.1 General Matrix Theory

Most of the theorems discussed are classical theorems and can be found in the

books [28], [31], and [32]. Let M be a square matrix where M = {mij}. We will write

M ≥ 0 when mij ≥ 0 for each i and j and likewise M > 0 when mij > 0 for each i and j .

Definition 17 (Primitive Matrix). A square non-negative matrix M is said to be primitive if

there exists a positive integer k such that Mk > 0.

Definition 18 (Vector Norm). A norm ∥ · ∥ in a vector space V over a field F is a function

∥ · ∥ : V → R that satisfies the following for all x , y ∈ V and c ∈ F :

1. ∥cx∥ = |c |∥x∥,2. ∥x∥ = 0 if and only if x = 0,3. ∥x∥ ≥ 0, and4. ∥x + y∥ ≤ ∥x∥+ ∥y∥ (Triangle Inequality).

Definition 19 (Induced Matrix Norm). A vector norm that is defined on Cm, induces a

matrix norm on Cm×n by

∥M∥ = sup∥x∥=0

∥Mx∥∥x∥

where M ∈ Cm×n.

Definition 20 (Spectrum). The set {λ1, · · · ,λn} of eigenvalues of a square matrix M is

called the spectrum of M.

Definition 21 (Spectral Radius). Let M be a square matrix with eigenvalues {λ1, · · · ,λn},

then the spectral radius ρ(M) is given by

ρ(M) = max1≤i≤n

|λi |.

Theorem A.4. Suppose M is a square matrix and ∥ · ∥ is an induced matrix norm, then

ρ(M) ≤ ∥M∥

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Proof. Suppose M has an eigenvalue λj with a corresponding eigenvector vj , so that

Mvj = λjvj . Then we have

∥M∥ = sup∥x∥=0

∥Mx∥∥x∥

≥ ∥Mvj∥∥vj∥

=∥λjvj∥∥vj∥

= |λj |.

Since this is true for all eigenvalues of M, then ∥M∥ ≥ ρ(M).

The following theorem can be found in [33], Corollary 5.6.14.

Theorem A.5 (Gelfand’s formula). Suppose M is a square matrix with real entries and

∥ · ∥ is a matrix norm, then

limn→∞

∥Mn∥1/n = ρ(M)

A.2 Spectral Radius Properties

These next set of definitions and theorems can be found by [29]. Let M be a finite

set of square matrices and ρ denote the spectral radius of a matrix, i.e. the magnitude of

the largest eigenvalue. Define

ρk(M ) = maxA1,...,Ak∈M

ρ(A1 · · ·Ak)1/k

Definition 22 (generalized spectral radius).

ρ(M ) = lim supk→∞

ρk(M )

Theorem A.6. Let M = {M1, · · · ,Mn} be a set of square matrices with non negative

entries. Let S be the matrix whose entries are the componentwise maximum of the

entries of the matrices M . Then

ρ(S)

n≤ ρ(M ) ≤ ρ(S).

Corollary 4. Let M = {M1, · · · ,Mn} be a set of square matrices with non negative

entries. Then 0≤ρ(M ) < ∞.

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BIOGRAPHICAL SKETCH

Jason Harrington was born in Louisville, Kentucky. He earned his B.A. at Western

Kentucky University where he met his beautiful wife Leslie. While working on this

dissertation at the University of Florida they had their first son Daniel. In addition to

mathematics, he also enjoys computer programming and reading.

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