Semigroups of M obius transformations Matthew Jacques Thursday 12 - - PowerPoint PPT Presentation

Semigroups of M¨

• bius transformations

Matthew Jacques Thursday 12th March 2015

• Joint work with Ian Short -

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 0 / 29

Introduction Contents

Contents

1 M¨

• bius transformations and hyperbolic geometry

■ M¨

• bius transformations and their action inside the unit ball

■ The hyperbolic metric

2 Semigroups of M¨

• bius transformations

■ Semigroups ■ Limit sets of M¨

• bius semigroups

■ Examples

3 Composition sequences

■ Escaping and converging composition sequences ■ Examples

4 A Theorem on convergence

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 1 / 29

Introduction Contents

Contents

1 M¨

• bius transformations and hyperbolic geometry

■ M¨

• bius transformations and their action inside the unit ball

■ The hyperbolic metric

2 Semigroups of M¨

• bius transformations

■ Semigroups ■ Limit sets of M¨

• bius semigroups

■ Examples

3 Composition sequences

■ Escaping and converging composition sequences ■ Examples

4 A Theorem on convergence

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 1 / 29

Introduction Contents

Contents

1 M¨

• bius transformations and hyperbolic geometry

■ M¨

• bius transformations and their action inside the unit ball

■ The hyperbolic metric

2 Semigroups of M¨

• bius transformations

■ Semigroups ■ Limit sets of M¨

• bius semigroups

■ Examples

3 Composition sequences

■ Escaping and converging composition sequences ■ Examples

4 A Theorem on convergence

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 1 / 29

Introduction Contents

Contents

1 M¨

• bius transformations and hyperbolic geometry

■ M¨

• bius transformations and their action inside the unit ball

■ The hyperbolic metric

2 Semigroups of M¨

• bius transformations

■ Semigroups ■ Limit sets of M¨

• bius semigroups

■ Examples

3 Composition sequences

■ Escaping and converging composition sequences ■ Examples

4 A Theorem on convergence

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 1 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

M¨

• bius transformations

M¨

• bius transformations are the conformal automorphisms of

❜

❈ = ❈ ❬ ❢✶❣. That is the bijective functions on ❜ ❈ which preserve angles and their

• rientation.

Each takes the form z ✼ ✦ az + b cz + d with a❀ b❀ c❀ d ✷ ❈ and ad bc ✻= 0

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 2 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

M¨

• bius transformations

M¨

• bius transformations are the conformal automorphisms of

❜

❈ = ❈ ❬ ❢✶❣. That is the bijective functions on ❜ ❈ which preserve angles and their

• rientation.

Each takes the form z ✼ ✦ az + b cz + d with a❀ b❀ c❀ d ✷ ❈ and ad bc ✻= 0

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 2 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

We consider the group ▼ of M¨

• bius transformations acting on ❜

❈, which we identify with ❙2. By decomposing the action of any given M¨

• bius transformation into a

composition of inversions in spheres orthogonal to ❙2, the action of ▼ may be extended to a conformal action on ❘3 ❬ ❢✶❣. In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 3 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

We consider the group ▼ of M¨

• bius transformations acting on ❜

❈, which we identify with ❙2. By decomposing the action of any given M¨

• bius transformation into a

composition of inversions in spheres orthogonal to ❙2, the action of ▼ may be extended to a conformal action on ❘3 ❬ ❢✶❣. In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 3 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

We consider the group ▼ of M¨

• bius transformations acting on ❜

❈, which we identify with ❙2. By decomposing the action of any given M¨

• bius transformation into a

composition of inversions in spheres orthogonal to ❙2, the action of ▼ may be extended to a conformal action on ❘3 ❬ ❢✶❣. In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 3 / 29

M¨

• bius transformations and hyperbolic geometry

M¨

• bius transformations

We consider the group ▼ of M¨

• bius transformations acting on ❜

❈, which we identify with ❙2. By decomposing the action of any given M¨

• bius transformation into a

composition of inversions in spheres orthogonal to ❙2, the action of ▼ may be extended to a conformal action on ❘3 ❬ ❢✶❣. In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 3 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

The hyperbolic metric, ✚(✁ ❀ ✁) on ❇3

The hyperbolic metric ✚ on ❇3 is induced by the infinitesimal metric ds = ❥dx❥ 1 ❥x❥2 ✿

• From any point inside ❇3 the distance to the ideal boundary, ❙2, is

infinite.

• Geodesics are circular arcs which when extended land orthogonally on

❙2. The group of M¨

• bius transformations that preserve ❇3 are exactly the set
• f orientation preserving isometries of (❇3❀ ✚).

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 4 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

The hyperbolic metric, ✚(✁ ❀ ✁) on ❇3

The hyperbolic metric ✚ on ❇3 is induced by the infinitesimal metric ds = ❥dx❥ 1 ❥x❥2 ✿

• From any point inside ❇3 the distance to the ideal boundary, ❙2, is

infinite.

• Geodesics are circular arcs which when extended land orthogonally on

❙2. The group of M¨

• bius transformations that preserve ❇3 are exactly the set
• f orientation preserving isometries of (❇3❀ ✚).

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 4 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

The hyperbolic metric, ✚(✁ ❀ ✁) on ❇3

The hyperbolic metric ✚ on ❇3 is induced by the infinitesimal metric ds = ❥dx❥ 1 ❥x❥2 ✿

• From any point inside ❇3 the distance to the ideal boundary, ❙2, is

infinite.

• Geodesics are circular arcs which when extended land orthogonally on

❙2. The group of M¨

• bius transformations that preserve ❇3 are exactly the set
• f orientation preserving isometries of (❇3❀ ✚).

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 4 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

The hyperbolic metric, ✚(✁ ❀ ✁) on ❇3

The hyperbolic metric ✚ on ❇3 is induced by the infinitesimal metric ds = ❥dx❥ 1 ❥x❥2 ✿

• From any point inside ❇3 the distance to the ideal boundary, ❙2, is

infinite.

• Geodesics are circular arcs which when extended land orthogonally on

❙2. The group of M¨

• bius transformations that preserve ❇3 are exactly the set
• f orientation preserving isometries of (❇3❀ ✚).

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 4 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 5 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 5 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 5 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

Aside from the identity, there are three types of M¨

• bius transformation.
• Loxodromic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ ✻= 1. Have two fixed points, one attracting and one repelling.

• Elliptic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ = 1. Have two neutral fixed points.

• Parabolic transformations

Conjugate to z ✼ ✦ z + 1. Have one neutral fixed point.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 6 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

Aside from the identity, there are three types of M¨

• bius transformation.
• Loxodromic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ ✻= 1. Have two fixed points, one attracting and one repelling.

• Elliptic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ = 1. Have two neutral fixed points.

• Parabolic transformations

Conjugate to z ✼ ✦ z + 1. Have one neutral fixed point.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 6 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

Aside from the identity, there are three types of M¨

• bius transformation.
• Loxodromic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ ✻= 1. Have two fixed points, one attracting and one repelling.

• Elliptic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ = 1. Have two neutral fixed points.

• Parabolic transformations

Conjugate to z ✼ ✦ z + 1. Have one neutral fixed point.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 6 / 29

M¨

• bius transformations and hyperbolic geometry

The hyperbolic metric

Aside from the identity, there are three types of M¨

• bius transformation.
• Loxodromic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ ✻= 1. Have two fixed points, one attracting and one repelling.

• Elliptic transformations

Conjugate to z ✼ ✦ ✕z where ❥✕❥ = 1. Have two neutral fixed points.

• Parabolic transformations

Conjugate to z ✼ ✦ z + 1. Have one neutral fixed point.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 6 / 29

Semigroups of M¨

• bius transformations

Semigroups

Definition

Given a set ❋ of M¨

• bius transformations, the semigroup S generated by ❋

is the set of finite (and non-empty) compositions of elements from ❋. We write S = ❤❋✐ as the semigroup generated by ❋.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 7 / 29

Semigroups of M¨

• bius transformations

Limit sets

Limit sets

Let S be a semigroup of M¨

• bius transformations.

Definition

The forwards limit set of S is the set Λ+(S) =

♥

z ✷ ❙2 ❥ lim

n✦✶ gn(✏) = z for some sequence gn in S

♦

✿ Similarly the backwards limit set of S is given by Λ(S) =

♥

z ✷ ❙2 ❥ lim

n✦✶ g1 n (✏) = z for some sequence gn in S

♦

✿ Since each gn is an isometry of the hyperbolic metric, these definitions are independent of the choice of ✏ ✷ ❇3.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 8 / 29

Semigroups of M¨

• bius transformations

Limit sets

Three characterisations

Write J(S) = subset of ❙2 upon which S is not a normal family.

Theorem D. Fried, S. Marotta and R. Stankewitz (2012)

For except for certain ”Elementary” semigroups, Λ(S) = J(S) = ❢Repelling fixed points of S❣ Λ+(S) = J(S1) = ❢Attracting fixed points of S❣✿

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 9 / 29

Semigroups of M¨

• bius transformations

Limit sets

Properties (Fried, Marotta and Stankewitz)

• Both Λ+, Λ are closed.
• Either ❥Λ+❥ ❁ 3 or Λ+ is a perfect set. Similarly for Λ.
• Λ+ is forward invariant under S, that is g(Λ+) ✒ Λ+ for all g ✷ S.
• If Λ+ contains at least two points then it is the smallest closed

forwards invariant set containing at least two points.

• Λ is backwards invariant under S, that is g1(Λ) ✒ Λ for all

g ✷ S.

• If Λ contains at least two points then it is the smallest closed

backwards invariant set containing at least two points.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 10 / 29

Semigroups of M¨

• bius transformations

Limit sets

Properties (Fried, Marotta and Stankewitz)

• Both Λ+, Λ are closed.
• Either ❥Λ+❥ ❁ 3 or Λ+ is a perfect set. Similarly for Λ.
• Λ+ is forward invariant under S, that is g(Λ+) ✒ Λ+ for all g ✷ S.
• If Λ+ contains at least two points then it is the smallest closed

forwards invariant set containing at least two points.

• Λ is backwards invariant under S, that is g1(Λ) ✒ Λ for all

g ✷ S.

• If Λ contains at least two points then it is the smallest closed

backwards invariant set containing at least two points.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 10 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of elementary semigroups

Elementary semigroups

❋ =

♥

z ✼ ✦ ei✒z

♦

Λ = Λ+ = ❀. ❋ = ❢z ✼ ✦ 2z❣ Λ = ❢0❣, Λ+ = ❢✶❣. ❋ = ❢z ✼ ✦ z + 1❣ Λ = Λ+ = ❢✶❣. ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Λ = ❢✶❣, Λ+ = middle thirds Cantor set.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 11 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Non-elementary Kleinian group

A Kleinian group is a group S such that the S orbit of any point in hyperbolic space is a discrete set of points. Any Kleinian group is a semigroup with equal forwards and backwards limit sets.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 12 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Non-elementary Kleinian group

A Kleinian group is a group S such that the S orbit of any point in hyperbolic space is a discrete set of points. Any Kleinian group is a semigroup with equal forwards and backwards limit sets.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 12 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Non-elementary Kleinian group

A Kleinian group is a group S such that the S orbit of any point in hyperbolic space is a discrete set of points. Any Kleinian group is a semigroup with equal forwards and backwards limit sets.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 12 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. f g Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. f g g1 f 1 Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. f g Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Subsemigroup of a Kleinian Group

Consider the Modular group Γ. Γ may be generated by two parabolic generators, f ❀ g. Λ+ Λ f g Let S be the semigroup generated by f ❀ g.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 13 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 14 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 15 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 16 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 17 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Schottky Semigroups

f S =

✡✟f ❀ g❀ h❀ h1✠☛

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 18 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Schottky Semigroups

f S =

✡✟f ❀ g❀ h❀ h1✠☛

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 18 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Schottky Semigroups

h h1 g f S =

✡✟f ❀ g❀ h❀ h1✠☛

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 18 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Schottky Semigroups

h h1 g f S =

✡✟f ❀ g❀ h❀ h1✠☛

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 18 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups

Schottky Semigroups

h h1 g f S =

✡✟f ❀ g❀ h❀ h1✠☛

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 18 / 29

Semigroups of M¨

• bius transformations

Examples of non-elementary semigroups Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 19 / 29

Composition sequences Composition sequences

Composition sequences

Fix a set of M¨

• bius transformations ❋.

A composition sequence of M¨

• bius transformations generated by ❋ is any

sequence with nth term Fn = f1 ✍ f2 ✍ ✁ ✁ ✁ ✍ fn❀ where each fi is chosen from ❋. Note the direction of composition.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 20 / 29

Composition sequences Composition sequences

Composition sequences

Fix a set of M¨

• bius transformations ❋.

A composition sequence of M¨

• bius transformations generated by ❋ is any

sequence with nth term Fn = f1 ✍ f2 ✍ ✁ ✁ ✁ ✍ fn❀ where each fi is chosen from ❋. Note the direction of composition.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 20 / 29

Composition sequences Composition sequences

Composition sequences

Fix a set of M¨

• bius transformations ❋.

A composition sequence of M¨

• bius transformations generated by ❋ is any

sequence with nth term Fn = f1 ✍ f2 ✍ ✁ ✁ ✁ ✍ fn❀ where each fi is chosen from ❋. Note the direction of composition.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 20 / 29

Composition sequences Composition sequences

Write f (z) = 1 z + 2 and g(z) = 3 z + 1✿ Then F1(z) = f (z) = 1 z + 2 F2(z) = f ✍ g(z) = 1 2 + 3 1 + z F3(z) = f ✍ g ✍ g(z) = 1 2 + 3 1 + 3 1 + z so that Fn(0) is the nth convergent of some continued fraction.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 21 / 29

Composition sequences Composition sequences

Write f (z) = 1 z + 2 and g(z) = 3 z + 1✿ Then F1(z) = f (z) = 1 z + 2 F2(z) = f ✍ g(z) = 1 2 + 3 1 + z F3(z) = f ✍ g ✍ g(z) = 1 2 + 3 1 + 3 1 + z so that Fn(0) is the nth convergent of some continued fraction.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 21 / 29

Composition sequences Composition sequences

Write f (z) = 1 z + 2 and g(z) = 3 z + 1✿ Then F1(z) = f (z) = 1 z + 2 F2(z) = f ✍ g(z) = 1 2 + 3 1 + z F3(z) = f ✍ g ✍ g(z) = 1 2 + 3 1 + 3 1 + z so that Fn(0) is the nth convergent of some continued fraction.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 21 / 29

Composition sequences Composition sequences

Write f (z) = 1 z + 2 and g(z) = 3 z + 1✿ Then F1(z) = f (z) = 1 z + 2 F2(z) = f ✍ g(z) = 1 2 + 3 1 + z F3(z) = f ✍ g ✍ g(z) = 1 2 + 3 1 + 3 1 + z so that Fn(0) is the nth convergent of some continued fraction.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 21 / 29

Composition sequences Composition sequences

Write f (z) = 1 z + 2 and g(z) = 3 z + 1✿ Then F1(z) = f (z) = 1 z + 2 F2(z) = f ✍ g(z) = 1 2 + 3 1 + z F3(z) = f ✍ g ✍ g(z) = 1 2 + 3 1 + 3 1 + z so that Fn(0) is the nth convergent of some continued fraction.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 21 / 29

Composition sequences Escaping and converging sequences

Escaping sequences

Definition

We say a sequence of M¨

• bius transformations gn is escaping if gn✏

accumulates only on the boundary of hyperbolic space. Equivalently ✚(gn✏❀ ✏) ✦ ✶ as n ✦ ✶✿

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 22 / 29

Composition sequences Escaping and converging sequences

Escaping sequences

Definition

We say a sequence of M¨

• bius transformations gn is escaping if gn✏

accumulates only on the boundary of hyperbolic space. Equivalently ✚(gn✏❀ ✏) ✦ ✶ as n ✦ ✶✿ gn✏ ✏

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 22 / 29

Composition sequences Escaping and converging sequences

Converging sequences

Definition

We say a sequence gn converges if gn✏ accumulates at exactly one point

• n the boundary of hyperbolic space.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 23 / 29

Composition sequences Escaping and converging sequences

Converging sequences

Definition

We say a sequence gn converges if gn✏ accumulates at exactly one point

• n the boundary of hyperbolic space.

gn✏ ✏

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 23 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

• ❋ =

♥

z ✼ ✦ 1

3z❀ z ✼

✦ 1

3z + 2 3

♦

Every composition sequence escapes? ✔ Every composition sequence converges? ✔

• ❋ such that ❋ generates a group.

Every composition sequence escapes? ✘ Every composition sequence converges? ✘

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 24 / 29

Composition sequences Escaping and converging sequences

Question:

Given a particular composition sequence, does it converge?

Related question:

Given a set of M¨

• bius transformations ❋ when does every composition

sequence generated by ❋

• escape,
• converge?

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 25 / 29

Composition sequences Escaping and converging sequences

Question:

Given a particular composition sequence, does it converge?

Related question:

Given a set of M¨

• bius transformations ❋ when does every composition

sequence generated by ❋

• escape,
• converge?

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 25 / 29

Main Theorem

Let S = ❤❋✐ be the semigroup generated by ❋.

Proposition

Every composition sequence generated by ❋ escapes if and only if Id ❂ ✷ S.

Proposition

If Λ+ and Λ are disjoint then every escaping composition sequence generated by ❋ converges. On the other hand:

Proposition

There is a dense G✍ set (w.r.t. the topology on Λ), D contained in Λ such that if Λ+ meets D, then not every composition sequence generated by ❋ converges.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 26 / 29

Main Theorem

Let S = ❤❋✐ be the semigroup generated by ❋.

Proposition

Every composition sequence generated by ❋ escapes if and only if Id ❂ ✷ S.

Proposition

If Λ+ and Λ are disjoint then every escaping composition sequence generated by ❋ converges. On the other hand:

Proposition

There is a dense G✍ set (w.r.t. the topology on Λ), D contained in Λ such that if Λ+ meets D, then not every composition sequence generated by ❋ converges.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 26 / 29

Main Theorem

Let S = ❤❋✐ be the semigroup generated by ❋.

Proposition

Every composition sequence generated by ❋ escapes if and only if Id ❂ ✷ S.

Proposition

If Λ+ and Λ are disjoint then every escaping composition sequence generated by ❋ converges. On the other hand:

Proposition

There is a dense G✍ set (w.r.t. the topology on Λ), D contained in Λ such that if Λ+ meets D, then not every composition sequence generated by ❋ converges.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 26 / 29

Main Theorem

Let S = ❤❋✐ be the semigroup generated by ❋.

Proposition

Every composition sequence generated by ❋ escapes if and only if Id ❂ ✷ S.

Proposition

If Λ+ and Λ are disjoint then every escaping composition sequence generated by ❋ converges. On the other hand:

Proposition

There is a dense G✍ set (w.r.t. the topology on Λ), D contained in Λ such that if Λ+ meets D, then not every composition sequence generated by ❋ converges.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 26 / 29

Main Theorem

Theorem

Suppose ❋ is bounded set of M¨

• bius transformations acting on ❇2,

generating a non-elementary semigroup S. Every composition sequence drawn from ❋ converges if and only if Id ❂ ✷ S and Λ+ is not the whole of ❙1.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 27 / 29

Main Theorem

Lemma

If S is a semigroup of M¨

• bius transformations acting on ❇3 such that

❥Λ❥ ❃ 1 and if Λ ✒ Λ+, then there exists a composition sequence in S that does not converge. Whenever Λ+ = ❙1 there exists some composition sequence that does not converge.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 28 / 29

Main Theorem

Lemma

If S is a semigroup of M¨

• bius transformations acting on ❇3 such that

❥Λ❥ ❃ 1 and if Λ ✒ Λ+, then there exists a composition sequence in S that does not converge. Whenever Λ+ = ❙1 there exists some composition sequence that does not converge.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 28 / 29

Main Theorem

Optimal?

Can we drop the reference to Λ+ = ❙1, in other words is the following true?

Conjecture

Suppose ❋ is bounded set of M¨

• bius transformations acting on ❇2,

generating a non-elementary semigroup S. Every composition sequence drawn from ❋ converges if and only if every composition sequence escapes.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 29 / 29

Literature

• B. Aebisher

The limiting behavior of sequences of M¨

• bius transformations

Mathematische Zeitschrift, 1990. Alan Beardon Continued Fractions, Discrete Groups and Complex Dynamics Computational Methods and Function Theory, 2001.

• D. Fried, S.M. Marotta and R. Stankewitz

Complex dynamics of M¨

• bius semigroups

Ergodic Theory Dynamical Systems, 2012.

• P. Mercat

Entropie des semi-groupes d’isomtrie d’un espace hyperbolique Preprint.

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 29 / 29

Thank you for your attention!

Matthew Jacques (The Open University) Semigroups of M¨

• bius transformations

Thursday 12th March 2015 29 / 29