Continued fractions and semigroups of

Möbius transformations

Ian Short

Wednesday 8 October 2014

Work in preparation

Matthew Jacques

Mairi Walker

A problem about continued fractions

Problem � version I. Determine those �nite sets of real

numbers X with the property that each continued fraction

1

b1 +1

b2 +1

b3 + � � �

with coe�cients in X converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Examples

Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.

Examples. X � N 4

X � R+ 4

X = f�1; 1g 8

X = f0g or X = fig 8

�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction

a1

b1 +a2

b2 +a3

b3 + � � �

converges.

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

Continued fractions and Möbius transformations

Tn = t1 � t2 � � � � � tn

A problem about sequences of Möbius transformations

De�nition. Given a set of (real) Möbius transformations F ,a composition sequence from F is a sequence

Fn = f1 � f2 � � � � � fn ; where fi 2 F :

Problem � version II. Determine those �nite sets of Möbius

transformations F with the property that every composition

sequence from F converges at 0.

A problem about sequences of Möbius transformations

De�nition. Given a set of (real) Möbius transformations F ,a composition sequence from F is a sequence

Fn = f1 � f2 � � � � � fn ; where fi 2 F :

Problem � version II. Determine those �nite sets of Möbius

transformations F with the property that every composition

sequence from F converges at 0.

Real Möbius transformations

z 7!az + b

cz + d; a ; b; c; d 2 R; ad � bc 6= 0

Real Möbius transformations

z 7!az + b

cz + d; a ; b; c; d 2 R; ad � bc 6= 0

Real Möbius transformations

z 7!az + b

cz + d; a ; b; c; d 2 R; ad � bc 6= 0

Real Möbius transformations

z 7!az + b

cz + d; a ; b; c; d 2 R; ad � bc 6= 0

Real Möbius transformations

z 7!az + b

cz + d; a ; b; c; d 2 R; ad � bc 6= 0

Conjugacy classi�cation of Möbius transformations

elliptic

one �xed pointinside

hyperbolic rotation

parabolic

one �xed pointon the boundary

z 7! z + 1

loxodromic

two �xed pointson the boundary

z 7! �z , � > 1

Conjugacy classi�cation of Möbius transformations

elliptic

one �xed pointinside

hyperbolic rotation

parabolic

one �xed pointon the boundary

z 7! z + 1

loxodromic

two �xed pointson the boundary

z 7! �z , � > 1

Conjugacy classi�cation of Möbius transformations

elliptic

one �xed pointinside

hyperbolic rotation

parabolic

one �xed pointon the boundary

z 7! z + 1

loxodromic

two �xed pointson the boundary

z 7! �z , � > 1

Example � the modular group

Problem � version II. Determine those �nite sets of Möbius

transformations F with the property that every composition

sequence from F converges at 0.

Example. 1 1

0 1

! 1 0

1 1

!

f (z ) = z + 1 g(z ) =z

z + 1

� =

�z 7!

az + b

cz + d: a ; b; c; d 2 Z; ad � bc = 1

�

Example � the modular group

Problem � version II. Determine those �nite sets of Möbius

transformations F with the property that every composition

sequence from F converges at 0.

Example. 1 1

0 1

! 1 0

1 1

!

f (z ) = z + 1 g(z ) =z

z + 1

� =

�z 7!

az + b

cz + d: a ; b; c; d 2 Z; ad � bc = 1

�

Example � the modular group

Problem � version II. Determine those �nite sets of Möbius

transformations F with the property that every composition

sequence from F converges at 0.

Example. 1 1

0 1

! 1 0

1 1

!

f (z ) = z + 1 g(z ) =z

z + 1

� =

�z 7!

az + b

cz + d: a ; b; c; d 2 Z; ad � bc = 1

�

Farey graph

Farey graph

Farey graph

Example � pairs of parabolics

Example. 1 �0 1

! 1 0

� 1

!

f (z ) = z + � g(z ) =z

�z + 1

Theorem. Let F = ff ; gg. Then every composition sequence

from F converges if and only if �� =2 (�4; 0].

Example � even-integer continued fractions.

f (z ) = z + 2 g(z ) =z

2z + 1

Example � pairs of parabolics

Example. 1 �0 1

! 1 0

� 1

!

f (z ) = z + � g(z ) =z

�z + 1

Theorem. Let F = ff ; gg. Then every composition sequence

from F converges if and only if �� =2 (�4; 0].

Example � even-integer continued fractions.

f (z ) = z + 2 g(z ) =z

2z + 1

Example � pairs of parabolics

Example. 1 �0 1

! 1 0

� 1

!

f (z ) = z + � g(z ) =z

�z + 1

Theorem. Let F = ff ; gg. Then every composition sequence

from F converges if and only if �� =2 (�4; 0].

Example � even-integer continued fractions.

f (z ) = z + 2 g(z ) =z

2z + 1

Farey tree

Farey tree

Farey tree

Escaping sequences and general convergence

De�nition. A sequence Fn of Möbius transformations is an escaping

sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).

Escaping sequences and general convergence

De�nition. A sequence Fn of Möbius transformations is an escaping

sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).

Escaping sequences and general convergence

De�nition. A sequence Fn of Möbius transformations is an escaping

sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.

An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).

Escaping sequences and general convergence

De�nition. A sequence Fn of Möbius transformations is an escaping

sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.

An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).

Escaping sequences and general convergence

De�nition. A sequence Fn of Möbius transformations is an escaping

sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).

Classical convergence implies general convergence

Backwards limit sets and conical limit sets

De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1

n(�).

Backwards limit sets and conical limit sets

De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1

n(�).

Backwards limit sets and conical limit sets

De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1

n(�).

Backwards limit sets and conical limit sets

De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1

n(�).

Return to the problem

Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

Theorem. The conjecture is true if F has order two.

Return to the problem

Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

Theorem. The conjecture is true if F has order two.

Return to the problem

Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

Theorem. The conjecture is true if F has order two.

Return to the problem

Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

Theorem. The conjecture is true if F has order two.

From Möbius transformations to semigroups

Lemma. Let F be a �nite set of Möbius transformations, and

let S be the semigroup generated by F .

The following are

equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

De�nition. A semigroup S is inverse free if no element of S

has an inverse in S . Equivalently, S is inverse free if it does not

contain the identity element.

From Möbius transformations to semigroups

Lemma. Let F be a �nite set of Möbius transformations, and

let S be the semigroup generated by F . The following are

equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

De�nition. A semigroup S is inverse free if no element of S

has an inverse in S . Equivalently, S is inverse free if it does not

contain the identity element.

From Möbius transformations to semigroups

Lemma. Let F be a �nite set of Möbius transformations, and

let S be the semigroup generated by F . The following are

equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

De�nition. A semigroup S is inverse free if no element of S

has an inverse in S .

Equivalently, S is inverse free if it does not

contain the identity element.

From Möbius transformations to semigroups

Lemma. Let F be a �nite set of Möbius transformations, and

let S be the semigroup generated by F . The following are

equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

De�nition. A semigroup S is inverse free if no element of S

has an inverse in S . Equivalently, S is inverse free if it does not

contain the identity element.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F .

Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Discrete semigroups

Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) the identity element does not belong to the closure of S .

Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:

(i) each composition sequence from F is an escaping sequence

(ii) each composition sequence from F converges generally

(iii) S is discrete and inverse free.

Problem � version IV. Classify the inverse-free discrete semigroups.

Recap

Problem � version I. Determine those �nite sets of real numbers Xwith the property that each continued fraction with coe�cients in X

converges.

Problem � version II. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges at 0.

Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.

Problem � version IV. Classify the inverse-free discrete semigroups.

Selected literature on Möbius semigroups

Some Moebius semigroups on the 2-sphereC.S. BallantineJ. Math. Mech., 1962

On certain semigroups of hyperbolic isometriesT. Jørgensen and K. SmithDuke Math. J., 1990

Complex dynamics of Möbius semigroupsD. Fried, S.M. Marotta and R. StankewitzErgodic Theory Dynam. Systems, 2012

Entropie des semi-groupes d'isométrie d'un espace hyperboliqueP. MercatTo be published

Exceptional semigroup

f (z ) = 2z g(z ) = 12z + 1

Two-generator Fuchsian groups literature

The classi�cation of discrete 2-generator subgroups of PSL(2; R)J.P. MatelskiIsrael J. Math., 1982

An algorithm for 2-generator Fuchsian groupsJ. Gilman and B. MaskitMichigan Math. J., 1991

Two-generator discrete subgroups of PSL(2;R)J. GilmanMem. Amer. Math. Soc., 1995

Two-generator Fuchsian groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky groups

Schottky semigroups

Schottky semigroups

Schottky semigroups

Schottky semigroups

Schottky semigroups

Schottky semigroups

Schottky semigroups

Reverse triangle inequality

Two-generator inverse-free discrete semigroups

elliptic elliptic

8

elliptic parabolic

8

elliptic loxodromic

8

parabolic parabolic

4

parabolic loxodromic

4

loxodromic loxodromic

4

Two-generator inverse-free discrete semigroups

elliptic elliptic 8

elliptic parabolic 8

elliptic loxodromic 8

parabolic parabolic

4

parabolic loxodromic

4

loxodromic loxodromic

4

Two-generator inverse-free discrete semigroups

elliptic elliptic 8

elliptic parabolic 8

elliptic loxodromic 8

parabolic parabolic 4

parabolic loxodromic 4

loxodromic loxodromic 4

Pairs of parabolics

Pairs of parabolics

�

�

�

�

Pairs of loxodromics

Pairs of loxodromics

�

�

�

�

�

�

Solution to Problem III

Theorem. A set F = ff ; gg of two Möbius transformations has

the property that every composition sequence from F converges

generally if and only if one of the following conditions is

satis�ed:

(i) f and g are parabolic and fg is not elliptic

(ii) one of f or g is loxodromic and the other is either

parabolic or loxodromic, and fgn and f ng are not elliptic

for any positive integer n .

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each continued fraction with coe�cients in X

converges.

�1

b + zjbj < 2 �

1

b + zjbj > 2

Theorem. A �nite set X has the property that every minus

continued fraction with coe�cients in X converges if and only

if X � (�1; 2] [ [2;+1).

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each minus continued fraction with coe�cients in

X converges.

�1

b + zjbj < 2 �

1

b + zjbj > 2

Theorem. A �nite set X has the property that every minus

continued fraction with coe�cients in X converges if and only

if X � (�1; 2] [ [2;+1).

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each minus continued fraction with coe�cients in

X converges.

�1

b + zjbj < 2 �

1

b + zjbj > 2

Theorem. A �nite set X has the property that every minus

continued fraction with coe�cients in X converges if and only

if X � (�1; 2] [ [2;+1).

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each minus continued fraction with coe�cients in

X converges.

�1

b + zjbj < 2 �

1

b + zjbj > 2

Theorem. A �nite set X has the property that every minus

continued fraction with coe�cients in X converges if and only

if X � (�1; 2] [ [2;+1).

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each continued fraction with coe�cients in X

converges.

b

c

bc 2 (�4; 0]

Theorem. A set fb; cg of real numbers has the property that

every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each continued fraction with coe�cients in X

converges.

b

c

bc 2 (�4; 0]

Theorem. A set fb; cg of real numbers has the property that

every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].

Return to the original problem

Problem � version I. Determine those �nite sets X with the

property that each continued fraction with coe�cients in X

converges.

b

c

bc 2 (�4; 0]

Theorem. A set fb; cg of real numbers has the property that

every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].

Examples

1

1+ z

1

2+ z

1

b + zb 2 N 1

b + zb 2 R

1

z

z + 1z

z + 1z + 2

z

�2z + 1

�

�

Examples

1

1+ z

1

2+ z

1

b + zb 2 N

1

b + zb 2 R

1

z

z + 1z

z + 1z + 2

z

�2z + 1

�

�

Examples

1

1+ z

1

2+ z

1

b + zb 2 N 1

b + zb 2 R

1

z

z + 1z

z + 1z + 2

z

�2z + 1

�

�

Examples

1

1+ z

1

2+ z

1

b + zb 2 N 1

b + zb 2 R

1

z

z + 1z

z + 1z + 2

z

�2z + 1

�

�

Examples

1

1+ z

1

2+ z

1

b + zb 2 N 1

b + zb 2 R

1

z

z + 1z

z + 1

z + 2z

�2z + 1

�

�

Examples

1

1+ z

1

2+ z

1

b + zb 2 N 1

b + zb 2 R

1

z

z + 1z

z + 1z + 2

z

�2z + 1

�

�

The unknown

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.

Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F

converges generally.

The unknown

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.

Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F

converges generally.

The unknown

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.

Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F

converges generally.

The unknown

Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.

More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.

Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F

converges generally.

Gallery1

1Created using lim by Curt McMullen

Gallery1

1Created using lim by Curt McMullen

Gallery1

1Created using lim by Curt McMullen

Gallery1

1Created using lim by Curt McMullen

Thank you for your attention.