Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Outline

1 Motivation

2 Thurston’s classification theorem

3 Definition and properties of pseudo-Anosovs

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Hyperbolic automorphisms of T2

A ∈ GL2(Z), no eigenvalues on unit circle.

• real eigenvalues λ, ±1/λ (|λ| > 1)

• eigenvectors vu, vs.

R2 is foliated by unstable leaves which are straight linesparallel to the eigenvector vu. A sends each unstable leaf toan unstable leaf, stretching it by λ

A ((x0, y0) + tvu) = A(x0, y0) + tλvu.

It is also foliated by stable leaves parallel to vs. A sendseach stable leaf to a stable leaf, contracting it by 1/λ.

These foliations descend to unstable/stable foliations on T2.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

A =

(2 11 1

)

evals3±

√5

2, evec slopes

−1∓√

52

.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Summary

For a hyperbolic automorphism A : T2 → T2 (linear Anosovdiffeomorphism):

• T2 is partitioned into unstable leaves. A sends unstableleaves to unstable leaves, expanding them uniformly bysome λ > 1. (Unstable foliation.)

• T2 is also partitioned into stable leaves. A sends stableleaves to stable leaves, contracting them uniformly by1/λ. (Stable foliation.)

• Every leaf winds densely around T2.

• At any point of T2, there are local coordinates in whichthe foliations look like this:

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Summary

For a hyperbolic automorphism A : T2 → T2 (linear Anosovdiffeomorphism):

• T2 is partitioned into unstable leaves. A sends unstableleaves to unstable leaves, expanding them uniformly bysome λ > 1. (Unstable foliation.)

• T2 is also partitioned into stable leaves. A sends stableleaves to stable leaves, contracting them uniformly by1/λ. (Stable foliation.)

• Every leaf winds densely around T2.

• At any point of T2, there are local coordinates in whichthe foliations look like this:

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Summary

For a hyperbolic automorphism A : T2 → T2 (linear Anosovdiffeomorphism):

• T2 is partitioned into unstable leaves. A sends unstableleaves to unstable leaves, expanding them uniformly bysome λ > 1. (Unstable foliation.)

• T2 is also partitioned into stable leaves. A sends stableleaves to stable leaves, contracting them uniformly by1/λ. (Stable foliation.)

• Every leaf winds densely around T2.

• At any point of T2, there are local coordinates in whichthe foliations look like this:

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Summary

For a hyperbolic automorphism A : T2 → T2 (linear Anosovdiffeomorphism):

• T2 is partitioned into unstable leaves. A sends unstableleaves to unstable leaves, expanding them uniformly bysome λ > 1. (Unstable foliation.)

• T2 is also partitioned into stable leaves. A sends stableleaves to stable leaves, contracting them uniformly by1/λ. (Stable foliation.)

• Every leaf winds densely around T2.

• At any point of T2, there are local coordinates in whichthe foliations look like this:

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs

Can we do similar things on other surfaces, e.g. the sphereor a genus two surface (“double torus”)?

• Impossible to foliate any other (orientable) surface sothat the foliations are “regular” everywhere.

• Except on the sphere, we can have singular foliations,locally like:

• Homeomorphisms which preserve such foliations,stretching unstable leaves and contracting stableleaves are called pseudo-Anosov.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs

Can we do similar things on other surfaces, e.g. the sphereor a genus two surface (“double torus”)?

• Impossible to foliate any other (orientable) surface sothat the foliations are “regular” everywhere.

• Except on the sphere, we can have singular foliations,locally like:

• Homeomorphisms which preserve such foliations,stretching unstable leaves and contracting stableleaves are called pseudo-Anosov.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs

Can we do similar things on other surfaces, e.g. the sphereor a genus two surface (“double torus”)?

• Impossible to foliate any other (orientable) surface sothat the foliations are “regular” everywhere.

• Except on the sphere, we can have singular foliations,locally like:

regular 3-prong 4-prong

• Homeomorphisms which preserve such foliations,stretching unstable leaves and contracting stableleaves are called pseudo-Anosov.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs

Can we do similar things on other surfaces, e.g. the sphereor a genus two surface (“double torus”)?

• Impossible to foliate any other (orientable) surface sothat the foliations are “regular” everywhere.

• Except on the sphere, we can have singular foliations,locally like:

regular 3-prong 4-prong

• Homeomorphisms which preserve such foliations,stretching unstable leaves and contracting stableleaves are called pseudo-Anosov.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs are interesting because:

• They have complicated dynamics.

• They have special features which make it possible tounderstand their dynamics well.

• They have the simplest dynamics in their isotopy class.That is, it’s impossible to destroy any of the dynamics ofa pseudo-Anosov by deforming it continuously.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs are interesting because:

• They have complicated dynamics.

• They have special features which make it possible tounderstand their dynamics well.

• They have the simplest dynamics in their isotopy class.That is, it’s impossible to destroy any of the dynamics ofa pseudo-Anosov by deforming it continuously.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Pseudo-Anosovs are interesting because:

• They have complicated dynamics.

• They have special features which make it possible tounderstand their dynamics well.

• They have the simplest dynamics in their isotopy class.That is, it’s impossible to destroy any of the dynamics ofa pseudo-Anosov by deforming it continuously.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Classification of surfaces

A surface is a compact metric space, in which every pointhas a neighbourhood homeomorphic to R2. (No boundarythis week.)

• Orientable or non-orientable.

• Orientable surfaces are classified by their genus g ≥ 0.(Every orientable surface is homeomorphic to the“standard” surface of genus g for some g.)

• (Similarly for non-orientable surfaces, take connectedsums of RP2 rather than T2.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Classification of surfaces

A surface is a compact metric space, in which every pointhas a neighbourhood homeomorphic to R2. (No boundarythis week.)

• Orientable or non-orientable.

• Orientable surfaces are classified by their genus g ≥ 0.(Every orientable surface is homeomorphic to the“standard” surface of genus g for some g.)

• (Similarly for non-orientable surfaces, take connectedsums of RP2 rather than T2.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Classification of surfaces

A surface is a compact metric space, in which every pointhas a neighbourhood homeomorphic to R2. (No boundarythis week.)

• Orientable or non-orientable.

• Orientable surfaces are classified by their genus g ≥ 0.(Every orientable surface is homeomorphic to the“standard” surface of genus g for some g.)

g = 0 g = 1 g = 2 g = 3

• (Similarly for non-orientable surfaces, take connectedsums of RP2 rather than T2.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Classification of surfaces

A surface is a compact metric space, in which every pointhas a neighbourhood homeomorphic to R2. (No boundarythis week.)

• Orientable or non-orientable.

• Orientable surfaces are classified by their genus g ≥ 0.(Every orientable surface is homeomorphic to the“standard” surface of genus g for some g.)

g = 0 g = 1 g = 2 g = 3

• (Similarly for non-orientable surfaces, take connectedsums of RP2 rather than T2.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Isotopy

Two homeomorphisms f , g : X → X of a surface X areisotopic if one can be deformed into the other continuouslythrough homeomorphisms.(There is a continuous F : X × [0, 1] → X such that if wedefine ft : X → X by ft(x) = F (x , t), then:

• f0 = f .

• f1 = g.

• Each ft is a homeomorphism.)

Isotopy is an equivalence relation on Homeo (X ),equivalence classes called isotopy classes.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Isotopy

Two homeomorphisms f , g : X → X of a surface X areisotopic if one can be deformed into the other continuouslythrough homeomorphisms.(There is a continuous F : X × [0, 1] → X such that if wedefine ft : X → X by ft(x) = F (x , t), then:

• f0 = f .

• f1 = g.

• Each ft is a homeomorphism.)

Isotopy is an equivalence relation on Homeo (X ),equivalence classes called isotopy classes.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Reducibility

A simple closed curve in a surface X is essential if it doesn’tbound a disk:

A homeomorphism f : X → X is reducible if it preserves afinite collection of mutually disjoint essential simple closedcurves.

(Then reduce f by cutting X along these curves.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Reducibility

A simple closed curve in a surface X is essential if it doesn’tbound a disk:

A homeomorphism f : X → X is reducible if it preserves afinite collection of mutually disjoint essential simple closedcurves.

(Then reduce f by cutting X along these curves.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Reducibility

A simple closed curve in a surface X is essential if it doesn’tbound a disk:

A homeomorphism f : X → X is reducible if it preserves afinite collection of mutually disjoint essential simple closedcurves.

(Then reduce f by cutting X along these curves.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Thurston’s classification (1979)

Let X be a surface, f : X → X a homeomorphism.

Then f is isotopic to a homeomorphism g : X → X which iseither:

a) Reducible, or

b) Finite order (gn = identity for some n), or

c) pseudo-Anosov.

(If f is isotopic to a pseudo-Anosov, then it isn’t also isotopicto a reducible or finite order homeomorphism.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Thurston’s classification (1979)

Let X be a surface, f : X → X a homeomorphism.

Then f is isotopic to a homeomorphism g : X → X which iseither:

a) Reducible, or

b) Finite order (gn = identity for some n), or

c) pseudo-Anosov.

(If f is isotopic to a pseudo-Anosov, then it isn’t also isotopicto a reducible or finite order homeomorphism.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Thurston’s classification (1979)

Let X be a surface, f : X → X a homeomorphism.

Then f is isotopic to a homeomorphism g : X → X which iseither:

a) Reducible, or

b) Finite order (gn = identity for some n), or

c) pseudo-Anosov.

(If f is isotopic to a pseudo-Anosov, then it isn’t also isotopicto a reducible or finite order homeomorphism.)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Foliations

A foliation F of a surface X is a partition of X into subsets(called leaves of F), such that each x ∈ X is either a regularpoint

or is a p-pronged singularity for some p ≥ 3.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Foliations

A foliation F of a surface X is a partition of X into subsets(called leaves of F), such that each x ∈ X is either a regularpoint

or is a p-pronged singularity for some p ≥ 3.

p = 3 p = 4

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Euler-Poincaré formula

Let F be a foliation of a surface X of genus g. Let ps be thenumber of prongs at a singularity s. Then∑

s

(ps − 2) = 4(g − 1).

Sphere 4(g − 1) = −4, no foliations exist.

Torus 4(g − 1) = 0, only non-singular foliations.

Genus 2 4(g − 1) = 4, so we can have:• One 6-prong, or• One 5-prong, one 3-prong, or• Two 4-prong, or• One 4-prong, two 3-prongs, or• Four 3-prongs.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Euler-Poincaré formula

Let F be a foliation of a surface X of genus g. Let ps be thenumber of prongs at a singularity s. Then∑

s

(ps − 2) = 4(g − 1).

Sphere 4(g − 1) = −4, no foliations exist.

Torus 4(g − 1) = 0, only non-singular foliations.

Genus 2 4(g − 1) = 4, so we can have:• One 6-prong, or• One 5-prong, one 3-prong, or• Two 4-prong, or• One 4-prong, two 3-prongs, or• Four 3-prongs.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Euler-Poincaré formula

Let F be a foliation of a surface X of genus g. Let ps be thenumber of prongs at a singularity s. Then∑

s

(ps − 2) = 4(g − 1).

Sphere 4(g − 1) = −4, no foliations exist.

Torus 4(g − 1) = 0, only non-singular foliations.

Genus 2 4(g − 1) = 4, so we can have:• One 6-prong, or• One 5-prong, one 3-prong, or• Two 4-prong, or• One 4-prong, two 3-prongs, or• Four 3-prongs.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Euler-Poincaré formula

Let F be a foliation of a surface X of genus g. Let ps be thenumber of prongs at a singularity s. Then∑

s

(ps − 2) = 4(g − 1).

Sphere 4(g − 1) = −4, no foliations exist.

Torus 4(g − 1) = 0, only non-singular foliations.

Genus 2 4(g − 1) = 4, so we can have:• One 6-prong, or• One 5-prong, one 3-prong, or• Two 4-prong, or• One 4-prong, two 3-prongs, or• Four 3-prongs.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Measured foliations

A measured foliation (F , µ) of X is a foliation F togetherwith a measure µ, which assigns a positive length µ(α) toeach arc α in X transverse to F , with the followingproperties:

• Additive: µ(α1 · α2) = µ(α1) + µ(α2).

• Holonomy invariant: µ(α1) = µ(α2).

• No atoms: “µ(α) → 0 as α → ·”.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Measured foliations

A measured foliation (F , µ) of X is a foliation F togetherwith a measure µ, which assigns a positive length µ(α) toeach arc α in X transverse to F , with the followingproperties:

• Additive: µ(α1 · α2) = µ(α1) + µ(α2).

α1

α2

• Holonomy invariant: µ(α1) = µ(α2).

• No atoms: “µ(α) → 0 as α → ·”.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Measured foliations

A measured foliation (F , µ) of X is a foliation F togetherwith a measure µ, which assigns a positive length µ(α) toeach arc α in X transverse to F , with the followingproperties:

• Additive: µ(α1 · α2) = µ(α1) + µ(α2).

α1

α2

• Holonomy invariant: µ(α1) = µ(α2).

α1

α2

• No atoms: “µ(α) → 0 as α → ·”.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Measured foliations

A measured foliation (F , µ) of X is a foliation F togetherwith a measure µ, which assigns a positive length µ(α) toeach arc α in X transverse to F , with the followingproperties:

• Additive: µ(α1 · α2) = µ(α1) + µ(α2).

α1

α2

• Holonomy invariant: µ(α1) = µ(α2).

α1

α2

• No atoms: “µ(α) → 0 as α → ·”.

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

pseudo-AnosovTwo measured foliations are transverse if at every x ∈ Xthere’s a chart in which they look like:

Given a homeomorphism f : X → X , we write f (F , µ) for themeasured foliation whose leaves are f -images of leaves ofF , and which assigns the measure µ(f−1(α)) to atransverse arc α.A homeomorphism f : X → X is pseudo-Anosov if thereexist a transverse pair of measured foliations (Fs, µs),(Fu, µu) and a number λ > 1 such that

f (Fs, µs) = (Fs, (1/λ)µs), and

f (Fu, µu) = (Fu, λµu).

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

pseudo-AnosovTwo measured foliations are transverse if at every x ∈ Xthere’s a chart in which they look like:

Given a homeomorphism f : X → X , we write f (F , µ) for themeasured foliation whose leaves are f -images of leaves ofF , and which assigns the measure µ(f−1(α)) to atransverse arc α.A homeomorphism f : X → X is pseudo-Anosov if thereexist a transverse pair of measured foliations (Fs, µs),(Fu, µu) and a number λ > 1 such that

f (Fs, µs) = (Fs, (1/λ)µs), and

f (Fu, µu) = (Fu, λµu).

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

pseudo-AnosovTwo measured foliations are transverse if at every x ∈ Xthere’s a chart in which they look like:

Given a homeomorphism f : X → X , we write f (F , µ) for themeasured foliation whose leaves are f -images of leaves ofF , and which assigns the measure µ(f−1(α)) to atransverse arc α.A homeomorphism f : X → X is pseudo-Anosov if thereexist a transverse pair of measured foliations (Fs, µs),(Fu, µu) and a number λ > 1 such that

f (Fs, µs) = (Fs, (1/λ)µs), and

f (Fu, µu) = (Fu, λµu).

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)

Introduction tothe dynamics

of surfacehomeos

Motivation

Thurston’sclassificationtheorem

Definition andproperties ofpseudo-Anosovs

Example

g = 2, λ ' 4.39 (λ4 − 7λ3 + 13λ2 − 7λ + 1 = 0).

One six-prong.

1 → 1 2 1 4

2 → 1 2 2 1 4

3 → 1 3 1 3 1 4

4 → 1 3 1 4

2 2 3 21 2 0 00 0 2 11 1 1 1

(Foliations orientable)