continued fractions and semigroups of möbius transformations
TRANSCRIPT
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Continued fractions and semigroups of
Möbius transformations
Ian Short
Wednesday 8 October 2014
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Work in preparation
Matthew Jacques
Mairi Walker
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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.
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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.
![Page 5: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/5.jpg)
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.
![Page 6: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/6.jpg)
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.
![Page 7: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/7.jpg)
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.
![Page 8: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/8.jpg)
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.
![Page 9: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/9.jpg)
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.
![Page 10: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/10.jpg)
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
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Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
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Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
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Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
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Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
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Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
![Page 16: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/16.jpg)
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.
![Page 17: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/17.jpg)
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.
![Page 18: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/18.jpg)
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
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Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
![Page 20: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/20.jpg)
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
![Page 21: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/21.jpg)
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
![Page 22: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/22.jpg)
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
![Page 23: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/23.jpg)
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
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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
![Page 25: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/25.jpg)
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
![Page 26: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/26.jpg)
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
�
![Page 27: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/27.jpg)
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
�
![Page 28: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/28.jpg)
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
�
![Page 29: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/29.jpg)
Farey graph
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Farey graph
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Farey graph
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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
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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
![Page 34: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/34.jpg)
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
![Page 35: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/35.jpg)
Farey tree
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Farey tree
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Farey tree
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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).
![Page 39: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/39.jpg)
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).
![Page 40: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/40.jpg)
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).
![Page 41: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/41.jpg)
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).
![Page 42: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/42.jpg)
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).
![Page 43: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/43.jpg)
Classical convergence implies general convergence
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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(�).
![Page 45: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/45.jpg)
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(�).
![Page 46: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/46.jpg)
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(�).
![Page 47: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/47.jpg)
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(�).
![Page 48: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/48.jpg)
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.
![Page 49: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/49.jpg)
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.
![Page 50: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/50.jpg)
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.
![Page 51: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/51.jpg)
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.
![Page 52: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/52.jpg)
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.
![Page 53: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/53.jpg)
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.
![Page 54: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/54.jpg)
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.
![Page 55: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/55.jpg)
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.
![Page 56: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/56.jpg)
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.
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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.
![Page 58: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/58.jpg)
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.
![Page 59: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/59.jpg)
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.
![Page 60: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/60.jpg)
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.
![Page 61: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/61.jpg)
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.
![Page 62: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/62.jpg)
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.
![Page 63: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/63.jpg)
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
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Exceptional semigroup
f (z ) = 2z g(z ) = 12z + 1
![Page 65: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/65.jpg)
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
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Two-generator Fuchsian groups
![Page 67: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/67.jpg)
Schottky groups
![Page 68: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/68.jpg)
Schottky groups
![Page 69: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/69.jpg)
Schottky groups
![Page 70: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/70.jpg)
Schottky groups
![Page 71: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/71.jpg)
Schottky groups
![Page 72: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/72.jpg)
Schottky groups
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Schottky groups
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Schottky groups
![Page 75: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/75.jpg)
Schottky groups
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Schottky groups
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Schottky groups
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Schottky groups
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Schottky semigroups
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Schottky semigroups
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Schottky semigroups
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Schottky semigroups
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Schottky semigroups
![Page 84: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/84.jpg)
Schottky semigroups
![Page 85: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/85.jpg)
Schottky semigroups
![Page 86: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/86.jpg)
Reverse triangle inequality
![Page 87: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/87.jpg)
Two-generator inverse-free discrete semigroups
elliptic elliptic
8
elliptic parabolic
8
elliptic loxodromic
8
parabolic parabolic
4
parabolic loxodromic
4
loxodromic loxodromic
4
![Page 88: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/88.jpg)
Two-generator inverse-free discrete semigroups
elliptic elliptic 8
elliptic parabolic 8
elliptic loxodromic 8
parabolic parabolic
4
parabolic loxodromic
4
loxodromic loxodromic
4
![Page 89: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/89.jpg)
Two-generator inverse-free discrete semigroups
elliptic elliptic 8
elliptic parabolic 8
elliptic loxodromic 8
parabolic parabolic 4
parabolic loxodromic 4
loxodromic loxodromic 4
![Page 90: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/90.jpg)
Pairs of parabolics
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Pairs of parabolics
�
�
�
�
![Page 92: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/92.jpg)
Pairs of loxodromics
![Page 93: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/93.jpg)
Pairs of loxodromics
�
�
�
�
�
�
![Page 94: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/94.jpg)
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 .
![Page 95: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/95.jpg)
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).
![Page 96: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/96.jpg)
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).
![Page 97: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/97.jpg)
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).
![Page 98: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/98.jpg)
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).
![Page 99: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/99.jpg)
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].
![Page 100: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/100.jpg)
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].
![Page 101: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/101.jpg)
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].
![Page 102: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/102.jpg)
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
�
�
![Page 103: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/103.jpg)
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
�
�
![Page 104: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/104.jpg)
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
�
�
![Page 105: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/105.jpg)
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
�
�
![Page 106: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/106.jpg)
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
�
�
![Page 107: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/107.jpg)
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
�
�
![Page 108: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/108.jpg)
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.
![Page 109: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/109.jpg)
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.
![Page 110: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/110.jpg)
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.
![Page 111: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/111.jpg)
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.
![Page 112: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/112.jpg)
Gallery1
1Created using lim by Curt McMullen
![Page 113: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/113.jpg)
Gallery1
1Created using lim by Curt McMullen
![Page 114: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/114.jpg)
Gallery1
1Created using lim by Curt McMullen
![Page 115: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/115.jpg)
Gallery1
1Created using lim by Curt McMullen
![Page 116: Continued fractions and semigroups of Möbius transformations](https://reader036.vdocuments.us/reader036/viewer/2022062604/62b4fdf477691d0f8e059ff3/html5/thumbnails/116.jpg)
Thank you for your attention.