MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of hyperbolic dynamical systems

Charles Walkden

April 12, 2013 In this lecture we use symbolic dynamics to model more general hyperbolic dynamical systems. We can then use thermodynamic formalism to prove ergodic-theoretic results about such systems.

In the last two lectures we studied thermodynamic formalism in the context of one-sided aperiodic shifts of finite type. In the last two lectures we studied thermodynamic formalism in the context of one-sided aperiodic shifts of finite type.

In this lecture we use symbolic dynamics to model more general hyperbolic dynamical systems. We can then use thermodynamic formalism to prove ergodic-theoretic results about such systems. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions: Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t. −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 Hyperbolic dynamical systems Let T be a C 1 diffeomorphism of a compact Riemannian manifold M. Let Tx M be the tangent space at x ∈ M. Let Dx T : Tx M → TTx M be the derivative of T . Idea: At each point x ∈ M, there are two directions:

I T contracts exponentially fast in one direction. I T expands exponentially fast in the other. Definition T : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t. ∀x ∈ M, there is a splitting

s u Tx M = Ex ⊕ Ex

into DT -invariant sub-bundles E s , E u s.t.

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s s Ex . W (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†)

(The convergence in (†) is necessarily exponentially fast.)

Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}.

E s , E u are called the stable and unstable sub-bundles, respectively. W s (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†)

(The convergence in (†) is necessarily exponentially fast.)

Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}.

E s , E u are called the stable and unstable sub-bundles, respectively. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s Ex . (The convergence in (†) is necessarily exponentially fast.)

Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}.

E s , E u are called the stable and unstable sub-bundles, respectively. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s s Ex . W (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†) Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}.

E s , E u are called the stable and unstable sub-bundles, respectively. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s s Ex . W (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†)

(The convergence in (†) is necessarily exponentially fast.) W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}.

E s , E u are called the stable and unstable sub-bundles, respectively. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s s Ex . W (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†)

(The convergence in (†) is necessarily exponentially fast.)

Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). E s , E u are called the stable and unstable sub-bundles, respectively. E s is tangent to the stable foliation W s . W s (x) is the stable manifold through x and is an immersed submanifold tangent to s s Ex . W (x) is characterised by

W s (x) = {y ∈ M | d(T nx, T ny) → 0 as n → ∞}. (†)

(The convergence in (†) is necessarily exponentially fast.)

Similarly, E u is tangent to the unstable foliation W u comprising of unstable manifolds W u(x). W u(x) is characterised by

W u(x) = {y ∈ M | d(T −nx, T −ny) → 0 as n → ∞}. 2 2 2 2 Let T : R /Z → R /Z be the cat map

 x   2 1   x  T = mod 1. y 1 1 y

 2 1  Note has two eigenvalues 1 1 √ √ 3 + 5 3 − 5 λ = > 1 λ = ∈ (0, 1) u 2 s 2 with corresponding eigenvectors ! ! 1 1 √ √ vu = −1+ 5 vs = −1− 5 . 2 2

Motivating example: the Cat Map  2 1  Note has two eigenvalues 1 1 √ √ 3 + 5 3 − 5 λ = > 1 λ = ∈ (0, 1) u 2 s 2 with corresponding eigenvectors ! ! 1 1 √ √ vu = −1+ 5 vs = −1− 5 . 2 2

Motivating example: the Cat Map

2 2 2 2 Let T : R /Z → R /Z be the cat map

 x   2 1   x  T = mod 1. y 1 1 y with corresponding eigenvectors ! ! 1 1 √ √ vu = −1+ 5 vs = −1− 5 . 2 2

Motivating example: the Cat Map

2 2 2 2 Let T : R /Z → R /Z be the cat map

 x   2 1   x  T = mod 1. y 1 1 y

 2 1  Note has two eigenvalues 1 1 √ √ 3 + 5 3 − 5 λ = > 1 λ = ∈ (0, 1) u 2 s 2 Motivating example: the Cat Map

2 2 2 2 Let T : R /Z → R /Z be the cat map

 x   2 1   x  T = mod 1. y 1 1 y

 2 1  Note has two eigenvalues 1 1 √ √ 3 + 5 3 − 5 λ = > 1 λ = ∈ (0, 1) u 2 s 2 with corresponding eigenvectors ! ! 1 1 √ √ vu = −1+ 5 vs = −1− 5 . 2 2 s u Then Ex is parallel to vs , and Ex is parallel to vu.

y

x s u Then Ex is parallel to vs , and Ex is parallel to vu.

Tx Dx T

TTx M

xx

Tx M s u Then Ex is parallel to vs , and Ex is parallel to vu.

Tx Dx T

TTx M

xx

Tx M s u Then Ex is parallel to vs , and Ex is parallel to vu.

Tx Dx T

TTx M

xx

Tx M s u Then Ex is parallel to vs , and Ex is parallel to vu.

Tx Dx T

TTx M

xx

Tx M √ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.

x Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x Open Question: Which manifolds support Anosov diffeomorphisms?

The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x The stable manifold W s (x) is tangent to E s (x). It is an immersed 1-dimensional sub-manifold.√ As vs = (1, (−1 − 5)/2) has rationally independent coefficients, s 2 2 W (x) is dense in R /Z .

x

Open Question: Which manifolds support Anosov diffeomorphisms? A hyperbolic toral automorphism can be thought of in the following way:

k k k R is an additive group and Z ⊂ R is a cocompact lattice (i.e. a k k discrete subgroup such that R /Z is compact). If A is a k × k k integer matrix with det A = ±1 then A is an automorphism of R k that preserves the lattice Z . More generally suppose N is a nilpotent Lie group, eg. matrices of the form  1 x z   0 1 y  . 0 0 1

(x, y, z ∈ R) and let Γ ⊂ N be the cocompact lattice where x, y, z ∈ Z.

A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism is Anosov. k k k R is an additive group and Z ⊂ R is a cocompact lattice (i.e. a k k discrete subgroup such that R /Z is compact). If A is a k × k k integer matrix with det A = ±1 then A is an automorphism of R k that preserves the lattice Z . More generally suppose N is a nilpotent Lie group, eg. matrices of the form  1 x z   0 1 y  . 0 0 1

(x, y, z ∈ R) and let Γ ⊂ N be the cocompact lattice where x, y, z ∈ Z.

A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism is Anosov. A hyperbolic toral automorphism can be thought of in the following way: More generally suppose N is a nilpotent Lie group, eg. matrices of the form  1 x z   0 1 y  . 0 0 1

(x, y, z ∈ R) and let Γ ⊂ N be the cocompact lattice where x, y, z ∈ Z.

A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism is Anosov. A hyperbolic toral automorphism can be thought of in the following way:

k k k R is an additive group and Z ⊂ R is a cocompact lattice (i.e. a k k discrete subgroup such that R /Z is compact). If A is a k × k k integer matrix with det A = ±1 then A is an automorphism of R k that preserves the lattice Z . and let Γ ⊂ N be the cocompact lattice where x, y, z ∈ Z.

A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism is Anosov. A hyperbolic toral automorphism can be thought of in the following way:

k k k R is an additive group and Z ⊂ R is a cocompact lattice (i.e. a k k discrete subgroup such that R /Z is compact). If A is a k × k k integer matrix with det A = ±1 then A is an automorphism of R k that preserves the lattice Z . More generally suppose N is a nilpotent Lie group, eg. matrices of the form  1 x z   0 1 y  . 0 0 1

(x, y, z ∈ R) A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism is Anosov. A hyperbolic toral automorphism can be thought of in the following way:

k k k R is an additive group and Z ⊂ R is a cocompact lattice (i.e. a k k discrete subgroup such that R /Z is compact). If A is a k × k k integer matrix with det A = ±1 then A is an automorphism of R k that preserves the lattice Z . More generally suppose N is a nilpotent Lie group, eg. matrices of the form  1 x z   0 1 y  . 0 0 1

(x, y, z ∈ R) and let Γ ⊂ N be the cocompact lattice where x, y, z ∈ Z. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms.

Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms. Let A ∈ Aut (N) be an automorphism of N such that A(Γ) = Γ and DA = derivative of A has no eigenvalues of modulus 1. Then A induces a map

T : N/Γ −→ N/Γ: gΓ 7−→ AgΓ.

N/Γ is called a nilmanifold. If N/Γ supports an Anosov diffeomorphism then dim N ≥ 6. The only known examples of Anosov diffeomorphisms are on k k I tori R /Z

I nilmanifolds N/Γ

I infranilmanifolds Moreover, all known Anosov diffeomorphisms are conjugate to the algebraic Anosov automorphisms. There are no Anosov diffeomorphisms on homotopy spheres, the Klein bottle, etc.

Conjecture k k If M supports an Anosov diffeomorphism then M is a torus R /Z , a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact lattice), or an infranilmanifold.

s u Instead of requiring Tx M to have a hyperbolic splitting Ex ⊕ Ex for all x ∈ M, we could require a hyperbolic splitting for all x in a T -invariant compact set Λ ⊂ M.

Topologically, Λ may be very complicated.

Locally maximal hyperbolic sets

T : M → M is Anosov if there is a hyperbolic splitting at all x ∈ M. This is a very strong condition on M. Conjecture k k If M supports an Anosov diffeomorphism then M is a torus R /Z , a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact lattice), or an infranilmanifold.

s u Instead of requiring Tx M to have a hyperbolic splitting Ex ⊕ Ex for all x ∈ M, we could require a hyperbolic splitting for all x in a T -invariant compact set Λ ⊂ M.

Topologically, Λ may be very complicated.

Locally maximal hyperbolic sets

T : M → M is Anosov if there is a hyperbolic splitting at all x ∈ M. This is a very strong condition on M.

There are no Anosov diffeomorphisms on homotopy spheres, the Klein bottle, etc. s u Instead of requiring Tx M to have a hyperbolic splitting Ex ⊕ Ex for all x ∈ M, we could require a hyperbolic splitting for all x in a T -invariant compact set Λ ⊂ M.

Topologically, Λ may be very complicated.

Locally maximal hyperbolic sets

T : M → M is Anosov if there is a hyperbolic splitting at all x ∈ M. This is a very strong condition on M.

There are no Anosov diffeomorphisms on homotopy spheres, the Klein bottle, etc.

Conjecture k k If M supports an Anosov diffeomorphism then M is a torus R /Z , a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact lattice), or an infranilmanifold. Topologically, Λ may be very complicated.

Locally maximal hyperbolic sets

T : M → M is Anosov if there is a hyperbolic splitting at all x ∈ M. This is a very strong condition on M.

There are no Anosov diffeomorphisms on homotopy spheres, the Klein bottle, etc.

Conjecture k k If M supports an Anosov diffeomorphism then M is a torus R /Z , a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact lattice), or an infranilmanifold.

s u Instead of requiring Tx M to have a hyperbolic splitting Ex ⊕ Ex for all x ∈ M, we could require a hyperbolic splitting for all x in a T -invariant compact set Λ ⊂ M. Locally maximal hyperbolic sets

T : M → M is Anosov if there is a hyperbolic splitting at all x ∈ M. This is a very strong condition on M.

There are no Anosov diffeomorphisms on homotopy spheres, the Klein bottle, etc.

Conjecture k k If M supports an Anosov diffeomorphism then M is a torus R /Z , a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact lattice), or an infranilmanifold.

s u Instead of requiring Tx M to have a hyperbolic splitting Ex ⊕ Ex for all x ∈ M, we could require a hyperbolic splitting for all x in a T -invariant compact set Λ ⊂ M.

Topologically, Λ may be very complicated. s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞ Anosov diffeomorphisms are attractors (with Λ = U = M).

Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Definition A compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic set (or basic set) if s u 1. there exists a splitting Tx M = Ex ⊕ Ex such that for all x ∈ Λ

n n s kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0 −n n u kDx T (v)k ≤ Cλ kvk ∀v ∈ Ex , ∀n ≥ 0.

2. T :Λ → Λ has a dense set of periodic points, has a dense orbit, and is not a single orbit. 3. ∃ an open set U ⊃ Λ s.t.

∞ \ T nU = Λ. n=−∞

T∞ n If in (3) we have n=0 T U = Λ, then Λ is an attractor. Anosov diffeomorphisms are attractors (with Λ = U = M). Define

T

If one iterates T forwards:

Example 1: The Smale horseshoe If one iterates T forwards:

Example 1: The Smale horseshoe

Define

T Example 1: The Smale horseshoe

Define

T

If one iterates T forwards: Example 1: The Smale horseshoe

Define

T

If one iterates T forwards: T −1

There is a locally maximal hyperbolic set

-the product of two Cantor sets.

and backwards There is a locally maximal hyperbolic set

-the product of two Cantor sets.

and backwards

T −1 -the product of two Cantor sets.

and backwards

T −1

There is a locally maximal hyperbolic set -the product of two Cantor sets.

and backwards

T −1

There is a locally maximal hyperbolic set and backwards

T −1

There is a locally maximal hyperbolic set

-the product of two Cantor sets. Let M = solid torus. Define a map T : M → M

Take a cross-section of the torus:

Example 2: The Solenoid Attractor Define a map T : M → M

Take a cross-section of the torus:

Example 2: The Solenoid Attractor

Let M = solid torus. Take a cross-section of the torus:

Example 2: The Solenoid Attractor

Let M = solid torus. Define a map T : M → M Take a cross-section of the torus:

Example 2: The Solenoid Attractor

Let M = solid torus. Define a map T : M → M Take a cross-section of the torus:

Example 2: The Solenoid Attractor

Let M = solid torus. Define a map T : M → M Example 2: The Solenoid Attractor

Let M = solid torus. Define a map T : M → M

Take a cross-section of the torus: Example 2: The Solenoid Attractor

Let M = solid torus. Define a map T : M → M

Take a cross-section of the torus: Open Question: Give a reasonable classification of all locally maximal hyperbolic sets.

(Conjecture: Are they all locally the product of a manifold and a Cantor set?)

T∞ n So Λ = n=0 T M is 1-dimensional and in each cross section is a Cantor set. (Conjecture: Are they all locally the product of a manifold and a Cantor set?)

T∞ n So Λ = n=0 T M is 1-dimensional and in each cross section is a Cantor set.

Open Question: Give a reasonable classification of all locally maximal hyperbolic sets. T∞ n So Λ = n=0 T M is 1-dimensional and in each cross section is a Cantor set.

Open Question: Give a reasonable classification of all locally maximal hyperbolic sets.

(Conjecture: Are they all locally the product of a manifold and a Cantor set?) Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} (It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.)

Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}. Symbolic dynamics and Markov Partitions

Let Λ be a locally maximal hyperbolic set and let T :Λ → Λ be hyperbolic. We want to code the dynamics of T by using an (aperiodic) shift of finite type. Let x ∈ Λ. Typically the stable and unstable manifolds will be dense in Λ. Define the local stable and unstable manifolds to be

s n n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0} u −n −n W (x) = {y ∈ M | d(T x, T y) ≤ , ∀n ≥ 0}.

(It follows that d(T nx, T ny) → 0 exponentially fast as n → ±∞ respectively.) u s [x, y] = W (x) ∩ W (y).

Thus [x, y] is a point whose orbit approximates that of y (in forward time) and approximates x (in backwards time).

If x, y ∈ Λ are sufficiently close then we define their “product” to be Thus [x, y] is a point whose orbit approximates that of y (in forward time) and approximates x (in backwards time).

If x, y ∈ Λ are sufficiently close then we define their “product” to be u s [x, y] = W (x) ∩ W (y). If x, y ∈ Λ are sufficiently close then we define their “product” to be u s [x, y] = W (x) ∩ W (y).

Thus [x, y] is a point whose orbit approximates that of y (in forward time) and approximates x (in backwards time). Example: The Cat Map

y

u W (y) x Example: The Cat Map

y

u W (x) x Example: The Cat Map

y

u W (x) x Example: The Cat Map

s W (y) y

u W (x) x Example: The Cat Map

s W (y) y

[x, y] u W (x) x If R is a rectangle and x ∈ R then we define

s s W (x, R) = W ∩ R u u W (x, R) = W ∩ R.

A set R ⊂ Λ is called a rectangle if

x, y sufficiently close =⇒ [x, y] ∈ R.

W u(x, R)

x

W s (x, R) A set R ⊂ Λ is called a rectangle if

x, y sufficiently close =⇒ [x, y] ∈ R.

If R is a rectangle and x ∈ R then we define

s s W (x, R) = W ∩ R u u W (x, R) = W ∩ R.

W u(x, R)

x

W s (x, R) A set R ⊂ Λ is called a rectangle if

x, y sufficiently close =⇒ [x, y] ∈ R.

If R is a rectangle and x ∈ R then we define

s s W (x, R) = W ∩ R u u W (x, R) = W ∩ R.

W u(x, R)

x

W s (x, R) A set R ⊂ Λ is called a rectangle if

x, y sufficiently close =⇒ [x, y] ∈ R.

If R is a rectangle and x ∈ R then we define

s s W (x, R) = W ∩ R u u W (x, R) = W ∩ R.

W u(x, R)

x

W s (x, R) A set R ⊂ Λ is called a rectangle if

x, y sufficiently close =⇒ [x, y] ∈ R.

If R is a rectangle and x ∈ R then we define

s s W (x, R) = W ∩ R u u W (x, R) = W ∩ R.

W u(x, R)

x

W s (x, R) T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 T (W u(x, R)) is a union of sets of the form W u(y, R0). =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if

x ∈ Int R

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 T (W u(x, R)) is a union of sets of the form W u(y, R0).

T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if

x ∈ Int R =⇒

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 A partition R = {R1,..., Rk } of Λ into rectangles is called a Markov partition if T (W u(x, R)) is a union of sets of the form W u(y, R0). x ∈ Int R =⇒ T −1(W s (y, R)) is a union of sets of the form W s (y, R0).

W u(x, R) TW u(x, R)

R x 4 R1 R3

R2 For ∞ n x ∈ ∩n=−∞T (Int R), we can code the orbit of x by recording the sequence of rectangles the orbit visits. Note that (cf decimal S∞ n expansions) if x ∈ n=−∞ T (∂R) then the coding is not unique.

Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4. Note that (cf decimal S∞ n expansions) if x ∈ n=−∞ T (∂R) then the coding is not unique.

Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4. For ∞ n x ∈ ∩n=−∞T (Int R), we can code the orbit of x by recording the sequence of rectangles the orbit visits. Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4. For ∞ n x ∈ ∩n=−∞T (Int R), we can code the orbit of x by recording the sequence of rectangles the orbit visits. Note that (cf decimal S∞ n expansions) if x ∈ n=−∞ T (∂R) then the coding is not unique. Let T be a hyperbolic map on a locally maximal hyperbolic set. Then there exists a Markov partition R = {R1,..., Rk } with an arbitrarily small diameter. In particular, there exists an aperiodic 2-sided shift of finite type Σ s.t. 1. the coding map ∞ ∞ \ −j π :Σ −→ Λ:(xj )−∞ 7−→ T Rxj −∞

is continuous, surjective, and injective except on S∞ −j −∞ T (∂R) σ Σ / Σ 2. π π commutes.   Λ / Λ T

Theorem (Bowen) In particular, there exists an aperiodic 2-sided shift of finite type Σ s.t. 1. the coding map ∞ ∞ \ −j π :Σ −→ Λ:(xj )−∞ 7−→ T Rxj −∞

is continuous, surjective, and injective except on S∞ −j −∞ T (∂R) σ Σ / Σ 2. π π commutes.   Λ / Λ T

Theorem (Bowen) Let T be a hyperbolic map on a locally maximal hyperbolic set. Then there exists a Markov partition R = {R1,..., Rk } with an arbitrarily small diameter. 1. the coding map ∞ ∞ \ −j π :Σ −→ Λ:(xj )−∞ 7−→ T Rxj −∞

is continuous, surjective, and injective except on S∞ −j −∞ T (∂R) σ Σ / Σ 2. π π commutes.   Λ / Λ T

Theorem (Bowen) Let T be a hyperbolic map on a locally maximal hyperbolic set. Then there exists a Markov partition R = {R1,..., Rk } with an arbitrarily small diameter. In particular, there exists an aperiodic 2-sided shift of finite type Σ s.t. σ Σ / Σ 2. π π commutes.   Λ / Λ T

Theorem (Bowen) Let T be a hyperbolic map on a locally maximal hyperbolic set. Then there exists a Markov partition R = {R1,..., Rk } with an arbitrarily small diameter. In particular, there exists an aperiodic 2-sided shift of finite type Σ s.t. 1. the coding map ∞ ∞ \ −j π :Σ −→ Λ:(xj )−∞ 7−→ T Rxj −∞

is continuous, surjective, and injective except on S∞ −j −∞ T (∂R) Theorem (Bowen) Let T be a hyperbolic map on a locally maximal hyperbolic set. Then there exists a Markov partition R = {R1,..., Rk } with an arbitrarily small diameter. In particular, there exists an aperiodic 2-sided shift of finite type Σ s.t. 1. the coding map ∞ ∞ \ −j π :Σ −→ Λ:(xj )−∞ 7−→ T Rxj −∞

is continuous, surjective, and injective except on S∞ −j −∞ T (∂R) σ Σ / Σ 2. π π commutes.   Λ / Λ T Example: Cat Map

This gives the matrix:

0 1 1 0 1 0 1 B 1 1 0 1 0 C B C B 1 1 0 1 0 C B C @ 0 0 1 0 1 A 0 0 1 0 1 Example: Cat Map

This gives the matrix:

0 1 1 0 1 0 1 B 1 1 0 1 0 C B C B 1 1 0 1 0 C B C @ 0 0 1 0 1 A 0 0 1 0 1 Example: Cat Map

This gives the matrix:

0 1 1 0 1 0 1 B 1 1 0 1 0 C B C B 1 1 0 1 0 C B C @ 0 0 1 0 1 A 0 0 1 0 1 Example: Cat Map

R4 R 3 This gives the matrix: R0 0 1 R1 1 1 0 1 0 B 1 1 0 1 0 C B C R2 B 1 1 0 1 0 C B C @ 0 0 1 0 1 A 0 0 1 0 1 Example: Cat Map

R4 R 3 This gives the matrix: R0 0 1 R1 1 1 0 1 0 B 1 1 0 1 0 C B C R2 B 1 1 0 1 0 C B C @ 0 0 1 0 1 A 0 0 1 0 1 Remark In general, rectangles may be (geometrically) very complicated. For Anosov automorphisms of a k-dimensional torus, k ≥ 3, the boundary of a Markov partition will typically be a fractal. In previous lectures, thermodynamic formalism was defined for 1-sided shifts of finite type.

Ergodic theory and hyperbolic dynamics

We want to use the thermodynamic formalism to study a hyperbolic map T :Λ → Λ. Note that T is invertible, so the symbolic model σ :Σ → Σ is 2-sided. Ergodic theory and hyperbolic dynamics

We want to use the thermodynamic formalism to study a hyperbolic map T :Λ → Λ. Note that T is invertible, so the symbolic model σ :Σ → Σ is 2-sided. In previous lectures, thermodynamic formalism was defined for 1-sided shifts of finite type. Ergodic theory and hyperbolic dynamics

We want to use the thermodynamic formalism to study a hyperbolic map T :Λ → Λ. Note that T is invertible, so the symbolic model σ :Σ → Σ is 2-sided. In previous lectures, thermodynamic formalism was defined for 1-sided shifts of finite type. Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future

Let Σ = two sided shift of finite type Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future

Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ ∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future

Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future

Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R.

Functions of the future

Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. Functions of the future

Let Σ = two sided shift of finite type |first disagreement| dθ = metric given by θ Fθ(Σ, R) = {functions f :Σ → R that are θ-H¨older}.

∞ ∞ If x = (xj )j=−∞ ∈ Σ then we think of (xj )j=0 as “the future” and 0 (xj )−∞ as “the past”. Note that if f ∈ F then, typically, f (x) will depend both on θ(Σ,R) the future and the past. If f only depends on future coordinates, i.e.

f (x) = f (x0, x1,... )

then f can be regarded as being defined on the one-sided shift + f :Σ → R. Cohomologous functions have the same dynamic behaviour: if f , g are cohomologous then

n−1 n−1 X X f (σj x) = g(σj x) + u(σnx) − u(x) j=0 j=0 n−1 X = g(σj x) + O(1). j=0

Theorem Let f ∈ Fθ(Σ, R). Then f is cohomologous to a function + g ∈ F 1 (Σ , ) that depends only on the future. θ 2 R

Cohomologous functions Recall: two functions f , g :Σ → R are cohomologous if ∃u s.t.

f = g + uσ − u. n−1 X = g(σj x) + O(1). j=0

Theorem Let f ∈ Fθ(Σ, R). Then f is cohomologous to a function + g ∈ F 1 (Σ , ) that depends only on the future. θ 2 R

Cohomologous functions Recall: two functions f , g :Σ → R are cohomologous if ∃u s.t.

f = g + uσ − u.

Cohomologous functions have the same dynamic behaviour: if f , g are cohomologous then

n−1 n−1 X X f (σj x) = g(σj x) + u(σnx) − u(x) j=0 j=0 Theorem Let f ∈ Fθ(Σ, R). Then f is cohomologous to a function + g ∈ F 1 (Σ , ) that depends only on the future. θ 2 R

Cohomologous functions Recall: two functions f , g :Σ → R are cohomologous if ∃u s.t.

f = g + uσ − u.

Cohomologous functions have the same dynamic behaviour: if f , g are cohomologous then

n−1 n−1 X X f (σj x) = g(σj x) + u(σnx) − u(x) j=0 j=0 n−1 X = g(σj x) + O(1). j=0 Cohomologous functions Recall: two functions f , g :Σ → R are cohomologous if ∃u s.t.

f = g + uσ − u.

Cohomologous functions have the same dynamic behaviour: if f , g are cohomologous then

n−1 n−1 X X f (σj x) = g(σj x) + u(σnx) − u(x) j=0 j=0 n−1 X = g(σj x) + O(1). j=0

Theorem Let f ∈ Fθ(Σ, R). Then f is cohomologous to a function + g ∈ F 1 (Σ , ) that depends only on the future. θ 2 R 1. start with a hyperbolic T :Λ → Λ 2. code the dynamics of T by a 2-sided shift of finite type Σ with coding map π :Σ → Λ ˆ 3. if f :Λ → R is H¨olderthen f = f π ∈ Fθ(Σ, R) for some θ ∈ (0, 1). + 4. replace fˆ by a cohomologous functiong ˆ ∈ F 1 (Σ , ) and θ 2 R apply thermodynamic formalism.

This allows us to: 2. code the dynamics of T by a 2-sided shift of finite type Σ with coding map π :Σ → Λ ˆ 3. if f :Λ → R is H¨olderthen f = f π ∈ Fθ(Σ, R) for some θ ∈ (0, 1). + 4. replace fˆ by a cohomologous functiong ˆ ∈ F 1 (Σ , ) and θ 2 R apply thermodynamic formalism.

This allows us to: 1. start with a hyperbolic T :Λ → Λ ˆ 3. if f :Λ → R is H¨olderthen f = f π ∈ Fθ(Σ, R) for some θ ∈ (0, 1). + 4. replace fˆ by a cohomologous functiong ˆ ∈ F 1 (Σ , ) and θ 2 R apply thermodynamic formalism.

This allows us to: 1. start with a hyperbolic T :Λ → Λ 2. code the dynamics of T by a 2-sided shift of finite type Σ with coding map π :Σ → Λ + 4. replace fˆ by a cohomologous functiong ˆ ∈ F 1 (Σ , ) and θ 2 R apply thermodynamic formalism.

This allows us to: 1. start with a hyperbolic T :Λ → Λ 2. code the dynamics of T by a 2-sided shift of finite type Σ with coding map π :Σ → Λ ˆ 3. if f :Λ → R is H¨olderthen f = f π ∈ Fθ(Σ, R) for some θ ∈ (0, 1). This allows us to: 1. start with a hyperbolic T :Λ → Λ 2. code the dynamics of T by a 2-sided shift of finite type Σ with coding map π :Σ → Λ ˆ 3. if f :Λ → R is H¨olderthen f = f π ∈ Fθ(Σ, R) for some θ ∈ (0, 1). + 4. replace fˆ by a cohomologous functiong ˆ ∈ F 1 (Σ , ) and θ 2 R apply thermodynamic formalism. Let f :Λ → R be H¨older:

|f (x) − f (y)| ≤ C d(x, y)α.

˜ + Let π :Σ → Λ. Then f π :Σ → R ∈ Fθ. Let f :Σ → R be ˜ cohomologous to f ◦ π. Let νf be the equilibrium state for (f ), a σ-invariant measure.

−1 Let µf = νf ◦ π . Then µf is called an equilibrium state for f and is a T -invariant measure.

Application 1: Existence of equilibrium states

Let T :Λ → Λ be C 1 hyperbolic diffeomorphism of a basic set Λ. ˜ + Let π :Σ → Λ. Then f π :Σ → R ∈ Fθ. Let f :Σ → R be ˜ cohomologous to f ◦ π. Let νf be the equilibrium state for (f ), a σ-invariant measure.

−1 Let µf = νf ◦ π . Then µf is called an equilibrium state for f and is a T -invariant measure.

Application 1: Existence of equilibrium states

Let T :Λ → Λ be C 1 hyperbolic diffeomorphism of a basic set Λ. Let f :Λ → R be H¨older:

|f (x) − f (y)| ≤ C d(x, y)α. −1 Let µf = νf ◦ π . Then µf is called an equilibrium state for f and is a T -invariant measure.

Application 1: Existence of equilibrium states

Let T :Λ → Λ be C 1 hyperbolic diffeomorphism of a basic set Λ. Let f :Λ → R be H¨older:

|f (x) − f (y)| ≤ C d(x, y)α.

˜ + Let π :Σ → Λ. Then f π :Σ → R ∈ Fθ. Let f :Σ → R be ˜ cohomologous to f ◦ π. Let νf be the equilibrium state for (f ), a σ-invariant measure. Application 1: Existence of equilibrium states

Let T :Λ → Λ be C 1 hyperbolic diffeomorphism of a basic set Λ. Let f :Λ → R be H¨older:

|f (x) − f (y)| ≤ C d(x, y)α.

˜ + Let π :Σ → Λ. Then f π :Σ → R ∈ Fθ. Let f :Σ → R be ˜ cohomologous to f ◦ π. Let νf be the equilibrium state for (f ), a σ-invariant measure.

−1 Let µf = νf ◦ π . Then µf is called an equilibrium state for f and is a T -invariant measure. Let T : X → X be a smooth diffeomorphism. Typically T does not preserve the volume m. Even if m is T -invariant, then it need not be ergodic. What can we say about

n−1 1 X lim f (T j x) n→∞ n j=0

for m-a.e. x ∈ X ?

Application 2: SRB and physical measures

Let X be a compact Riemannian manifold equipped with the Riemannian volume m. Application 2: SRB and physical measures

Let X be a compact Riemannian manifold equipped with the Riemannian volume m. Let T : X → X be a smooth diffeomorphism. Typically T does not preserve the volume m. Even if m is T -invariant, then it need not be ergodic. What can we say about

n−1 1 X lim f (T j x) n→∞ n j=0

for m-a.e. x ∈ X ? Let µ be an ergodic measure. As X is compact, we know that C(X , R) is separable. It follows easily from Birhoff’s Ergodic Theorem that ∃N, µ(N) = 1 s.t.

n−1 1 X Z f (T j x) −→ f dµ ∀x ∈ N. n j=0

(i.e. the set of full measure for which the ergodic sums of continuous observables converges can be chosen to be independent of the observables).

Let T : X → X be a smooth diffeomorphism of a compact Riemannian manifold X . As X is compact, we know that C(X , R) is separable. It follows easily from Birhoff’s Ergodic Theorem that ∃N, µ(N) = 1 s.t.

n−1 1 X Z f (T j x) −→ f dµ ∀x ∈ N. n j=0

(i.e. the set of full measure for which the ergodic sums of continuous observables converges can be chosen to be independent of the observables).

Let T : X → X be a smooth diffeomorphism of a compact Riemannian manifold X . Let µ be an ergodic measure. It follows easily from Birhoff’s Ergodic Theorem that ∃N, µ(N) = 1 s.t.

n−1 1 X Z f (T j x) −→ f dµ ∀x ∈ N. n j=0

(i.e. the set of full measure for which the ergodic sums of continuous observables converges can be chosen to be independent of the observables).

Let T : X → X be a smooth diffeomorphism of a compact Riemannian manifold X . Let µ be an ergodic measure. As X is compact, we know that C(X , R) is separable. (i.e. the set of full measure for which the ergodic sums of continuous observables converges can be chosen to be independent of the observables).

Let T : X → X be a smooth diffeomorphism of a compact Riemannian manifold X . Let µ be an ergodic measure. As X is compact, we know that C(X , R) is separable. It follows easily from Birhoff’s Ergodic Theorem that ∃N, µ(N) = 1 s.t.

n−1 1 X Z f (T j x) −→ f dµ ∀x ∈ N. n j=0 Let T : X → X be a smooth diffeomorphism of a compact Riemannian manifold X . Let µ be an ergodic measure. As X is compact, we know that C(X , R) is separable. It follows easily from Birhoff’s Ergodic Theorem that ∃N, µ(N) = 1 s.t.

n−1 1 X Z f (T j x) −→ f dµ ∀x ∈ N. n j=0

(i.e. the set of full measure for which the ergodic sums of continuous observables converges can be chosen to be independent of the observables). (We know B(Λ) contains an open set, so has positive Riemannian volume.)

Idea: We think of m-a.e. point as being ‘typical’, in the sense that m is a naturally occurring measure. Question: What happens to ergodic sums of continuous ob- servables for m-a.e. point? 1 Pn−1 j i.e. does limn→∞ n j=0 f (T x) exist for all con- tinuous f , m-a.e., and what is the limit?

Suppose T : X → X contains a locally maximal attractor, T :Λ → Λ (not necessarily hyperbolic). The basin of attraction B(Λ) is the set of points that converge under forward iteration to Λ. We think of m-a.e. point as being ‘typical’, in the sense that m is a naturally occurring measure. Question: What happens to ergodic sums of continuous ob- servables for m-a.e. point? 1 Pn−1 j i.e. does limn→∞ n j=0 f (T x) exist for all con- tinuous f , m-a.e., and what is the limit?

Suppose T : X → X contains a locally maximal attractor, T :Λ → Λ (not necessarily hyperbolic). The basin of attraction B(Λ) is the set of points that converge under forward iteration to Λ. (We know B(Λ) contains an open set, so has positive Riemannian volume.)

Idea: Question: What happens to ergodic sums of continuous ob- servables for m-a.e. point? 1 Pn−1 j i.e. does limn→∞ n j=0 f (T x) exist for all con- tinuous f , m-a.e., and what is the limit?

Suppose T : X → X contains a locally maximal attractor, T :Λ → Λ (not necessarily hyperbolic). The basin of attraction B(Λ) is the set of points that converge under forward iteration to Λ. (We know B(Λ) contains an open set, so has positive Riemannian volume.)

Idea: We think of m-a.e. point as being ‘typical’, in the sense that m is a naturally occurring measure. What happens to ergodic sums of continuous ob- servables for m-a.e. point? 1 Pn−1 j i.e. does limn→∞ n j=0 f (T x) exist for all con- tinuous f , m-a.e., and what is the limit?

Suppose T : X → X contains a locally maximal attractor, T :Λ → Λ (not necessarily hyperbolic). The basin of attraction B(Λ) is the set of points that converge under forward iteration to Λ. (We know B(Λ) contains an open set, so has positive Riemannian volume.)

Idea: We think of m-a.e. point as being ‘typical’, in the sense that m is a naturally occurring measure. Question: Suppose T : X → X contains a locally maximal attractor, T :Λ → Λ (not necessarily hyperbolic). The basin of attraction B(Λ) is the set of points that converge under forward iteration to Λ. (We know B(Λ) contains an open set, so has positive Riemannian volume.)

Idea: We think of m-a.e. point as being ‘typical’, in the sense that m is a naturally occurring measure. Question: What happens to ergodic sums of continuous ob- servables for m-a.e. point? 1 Pn−1 j i.e. does limn→∞ n j=0 f (T x) exist for all con- tinuous f , m-a.e., and what is the limit? 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r

S r 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r x r

S r 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r x r ~

S u r 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r x r ~

u S u r 2 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r x r ~ K

u S u r 2 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N

r x r ~ K

Tx r u S u r 2 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N T (N) = N r x r ~ K

Tx r u S u r 2 Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N T (N) = N r x r ~ K

Tx r u S u r 2 Then S is an attractor with basin S1 \{N}. As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0

Example: The North-South Map T : S 1 → S 1

N T (N) = N r x r ~ K

Tx r u S u r 2 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). Example: The North-South Map T : S 1 → S 1

N T (N) = N r x r ~ K

Tx r u S u r 2 1 Then S is an attractor with basin S \{N}. Let f ∈ C(X , R). As T nx → S ∀x ∈ S1 \{N}, we have

n−1 1 X Z f (T j x) −→ f (S) = f dδ . n S j=0 i.e. the measure we “see” by taking ergodic averages of m-a.e. point is the SRB measure.

The SRB measure is supported on the attractor Λ. As Λ may be (topologically) small, it may have zero Riemannian volume. (Example: the solenoid has zero volume.) Hence the SRB measure may be very different to the volume.

Definition Let T :Λ → Λ be an attractor. A T -invariant probability measure µ is an SRB (Sinai-Ruelle-Bowen) measure if

n−1 1 X Z f (T j x) −→ f dµ n j=0 almost everywhere on B(Λ) w.r.t. the Riemannian volume m. The SRB measure is supported on the attractor Λ. As Λ may be (topologically) small, it may have zero Riemannian volume. (Example: the solenoid has zero volume.) Hence the SRB measure may be very different to the volume.

Definition Let T :Λ → Λ be an attractor. A T -invariant probability measure µ is an SRB (Sinai-Ruelle-Bowen) measure if

n−1 1 X Z f (T j x) −→ f dµ n j=0 almost everywhere on B(Λ) w.r.t. the Riemannian volume m. i.e. the measure we “see” by taking ergodic averages of m-a.e. point is the SRB measure. As Λ may be (topologically) small, it may have zero Riemannian volume. (Example: the solenoid has zero volume.) Hence the SRB measure may be very different to the volume.

Definition Let T :Λ → Λ be an attractor. A T -invariant probability measure µ is an SRB (Sinai-Ruelle-Bowen) measure if

n−1 1 X Z f (T j x) −→ f dµ n j=0 almost everywhere on B(Λ) w.r.t. the Riemannian volume m. i.e. the measure we “see” by taking ergodic averages of m-a.e. point is the SRB measure.

The SRB measure is supported on the attractor Λ. Definition Let T :Λ → Λ be an attractor. A T -invariant probability measure µ is an SRB (Sinai-Ruelle-Bowen) measure if

n−1 1 X Z f (T j x) −→ f dµ n j=0 almost everywhere on B(Λ) w.r.t. the Riemannian volume m. i.e. the measure we “see” by taking ergodic averages of m-a.e. point is the SRB measure.

The SRB measure is supported on the attractor Λ. As Λ may be (topologically) small, it may have zero Riemannian volume. (Example: the solenoid has zero volume.) Hence the SRB measure may be very different to the volume. Remark Suppose T is an Anosov diffeomorphism and preserves volume. (Example, the cat map preserves Lebesgue measure = volume.) Then volume is the SRB measure. For a generic Anosov diffeomorphism, the SRB measure is not equal to volume.

Theorem Let T :Λ → Λ be a C 1+α hyperbolic attractor. Then there is a unique SRB measure. Moreover, it corresponds to the invariant Gibbs measure with potential − log dT |E u Then volume is the SRB measure. For a generic Anosov diffeomorphism, the SRB measure is not equal to volume.

Theorem Let T :Λ → Λ be a C 1+α hyperbolic attractor. Then there is a unique SRB measure. Moreover, it corresponds to the invariant Gibbs measure with potential − log dT |E u Remark Suppose T is an Anosov diffeomorphism and preserves volume. (Example, the cat map preserves Lebesgue measure = volume.) For a generic Anosov diffeomorphism, the SRB measure is not equal to volume.

Theorem Let T :Λ → Λ be a C 1+α hyperbolic attractor. Then there is a unique SRB measure. Moreover, it corresponds to the invariant Gibbs measure with potential − log dT |E u Remark Suppose T is an Anosov diffeomorphism and preserves volume. (Example, the cat map preserves Lebesgue measure = volume.) Then volume is the SRB measure. Theorem Let T :Λ → Λ be a C 1+α hyperbolic attractor. Then there is a unique SRB measure. Moreover, it corresponds to the invariant Gibbs measure with potential − log dT |E u Remark Suppose T is an Anosov diffeomorphism and preserves volume. (Example, the cat map preserves Lebesgue measure = volume.) Then volume is the SRB measure. For a generic Anosov diffeomorphism, the SRB measure is not equal to volume. Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type,

I ...

(Not) the end We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type,

I ...

(Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type,

I ...

(Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study: I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type,

I ...

(Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows, I thermodynamic formalism for countable state shifts of finite type,

I ...

(Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u, I ...

(Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type, (Not) the end

Ergodic theory is a huge subject with many connections to other areas of mathematics. The material in this course reflects my own interests. We could go on to study:

I the geodesic flow on negatively curved manifolds—these are hyperbolic flows,

I partially hyperbolic dynamical systems—where the tangent bundle splits into 3 invariant sub-bundles E s ⊕ E c ⊕ E u,

I thermodynamic formalism for countable state shifts of finite type,

I ...