magic: ergodic theory lecture 10 - the ergodic theory of ... · ergodic theory of hyperbolic...
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![Page 1: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/1.jpg)
MAGIC: Ergodic Theory Lecture 10 - The
ergodic theory of hyperbolic dynamical systems
Charles Walkden
April 12, 2013
![Page 2: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/2.jpg)
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.
![Page 3: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/3.jpg)
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.
![Page 4: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/4.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 5: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/5.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 6: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/6.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 7: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/7.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 8: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/8.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 9: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/9.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 10: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/10.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 11: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/11.jpg)
Hyperbolic dynamical systemsLet T be a C 1 diffeomorphism of a compact Riemannian manifold
M.
Let TxM be the tangent space at x ∈ M.
Let DxT : TxM → TTxM 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.
DefinitionT : M → M is an Anosov diffeomorphism if ∃C > 0, λ ∈ (0, 1) s.t.
∀x ∈ M, there is a splitting
TxM = E sx ⊕ Eu
x
into DT -invariant sub-bundles E s , Eu s.t.
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀n ≥ 0.
![Page 12: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/12.jpg)
E s , Eu 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
E sx . 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, Eu 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→∞}.
![Page 13: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/13.jpg)
E s , Eu 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
E sx .
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, Eu 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→∞}.
![Page 14: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/14.jpg)
E s , Eu 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
E sx . 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, Eu 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→∞}.
![Page 15: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/15.jpg)
E s , Eu 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
E sx . 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, Eu 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→∞}.
![Page 16: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/16.jpg)
E s , Eu 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
E sx . 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, Eu 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→∞}.
![Page 17: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/17.jpg)
E s , Eu 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
E sx . 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, Eu 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→∞}.
![Page 18: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/18.jpg)
Motivating example: the Cat Map
Let T : R2/Z2 → R2/Z2 be the cat map
T
(x
y
)=
(2 1
1 1
)(x
y
)mod 1.
Note
(2 1
1 1
)has two eigenvalues
λu =3 +√
5
2> 1 λs =
3−√
5
2∈ (0, 1)
with corresponding eigenvectors
vu =
(1
−1+√
52
)vs =
(1
−1−√
52
).
![Page 19: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/19.jpg)
Motivating example: the Cat Map
Let T : R2/Z2 → R2/Z2 be the cat map
T
(x
y
)=
(2 1
1 1
)(x
y
)mod 1.
Note
(2 1
1 1
)has two eigenvalues
λu =3 +√
5
2> 1 λs =
3−√
5
2∈ (0, 1)
with corresponding eigenvectors
vu =
(1
−1+√
52
)vs =
(1
−1−√
52
).
![Page 20: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/20.jpg)
Motivating example: the Cat Map
Let T : R2/Z2 → R2/Z2 be the cat map
T
(x
y
)=
(2 1
1 1
)(x
y
)mod 1.
Note
(2 1
1 1
)has two eigenvalues
λu =3 +√
5
2> 1 λs =
3−√
5
2∈ (0, 1)
with corresponding eigenvectors
vu =
(1
−1+√
52
)vs =
(1
−1−√
52
).
![Page 21: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/21.jpg)
Motivating example: the Cat Map
Let T : R2/Z2 → R2/Z2 be the cat map
T
(x
y
)=
(2 1
1 1
)(x
y
)mod 1.
Note
(2 1
1 1
)has two eigenvalues
λu =3 +√
5
2> 1 λs =
3−√
5
2∈ (0, 1)
with corresponding eigenvectors
vu =
(1
−1+√
52
)vs =
(1
−1−√
52
).
![Page 22: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/22.jpg)
Then E sx is parallel to vs , and Eu
x is parallel to vu.
y
x
![Page 23: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/23.jpg)
Then E sx is parallel to vs , and Eu
x is parallel to vu.
DxT
TxM
TTxM
xx
Tx
![Page 24: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/24.jpg)
Then E sx is parallel to vs , and Eu
x is parallel to vu.
DxT
TxM
TTxM
xx
Tx
![Page 25: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/25.jpg)
Then E sx is parallel to vs , and Eu
x is parallel to vu.
DxT
TxM
TTxM
xx
Tx
![Page 26: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/26.jpg)
Then E sx is parallel to vs , and Eu
x is parallel to vu.
DxT
TxM
TTxM
xx
Tx
![Page 27: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/27.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 28: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/28.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 29: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/29.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 30: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/30.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 31: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/31.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 32: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/32.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 33: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/33.jpg)
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,
W s(x) is dense in R2/Z2.
x
Open Question: Which manifolds support Anosov
diffeomorphisms?
![Page 34: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/34.jpg)
A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism
is Anosov.
A hyperbolic toral automorphism can be thought of in the
following way:
Rk is an additive group and Zk ⊂ Rk is a cocompact lattice (i.e. a
discrete subgroup such that Rk/Zk is compact). If A is a k × k
integer matrix with det A = ±1 then A is an automorphism of Rk
that preserves the lattice Zk .
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.
![Page 35: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/35.jpg)
A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism
is Anosov.
A hyperbolic toral automorphism can be thought of in the
following way:
Rk is an additive group and Zk ⊂ Rk is a cocompact lattice (i.e. a
discrete subgroup such that Rk/Zk is compact). If A is a k × k
integer matrix with det A = ±1 then A is an automorphism of Rk
that preserves the lattice Zk .
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.
![Page 36: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/36.jpg)
A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism
is Anosov.
A hyperbolic toral automorphism can be thought of in the
following way:
Rk is an additive group and Zk ⊂ Rk is a cocompact lattice (i.e. a
discrete subgroup such that Rk/Zk is compact). If A is a k × k
integer matrix with det A = ±1 then A is an automorphism of Rk
that preserves the lattice Zk .
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.
![Page 37: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/37.jpg)
A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism
is Anosov.
A hyperbolic toral automorphism can be thought of in the
following way:
Rk is an additive group and Zk ⊂ Rk is a cocompact lattice (i.e. a
discrete subgroup such that Rk/Zk is compact). If A is a k × k
integer matrix with det A = ±1 then A is an automorphism of Rk
that preserves the lattice Zk .
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.
![Page 38: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/38.jpg)
A hyperbolic ( = no eigenvalues of modulus 1) toral automorphism
is Anosov.
A hyperbolic toral automorphism can be thought of in the
following way:
Rk is an additive group and Zk ⊂ Rk is a cocompact lattice (i.e. a
discrete subgroup such that Rk/Zk is compact). If A is a k × k
integer matrix with det A = ±1 then A is an automorphism of Rk
that preserves the lattice Zk .
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.
![Page 39: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/39.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 40: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/40.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 41: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/41.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 42: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/42.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 43: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/43.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 44: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/44.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 45: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/45.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 46: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/46.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 47: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/47.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 48: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/48.jpg)
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 tori Rk/Zk
I nilmanifolds N/Γ
I infranilmanifolds
Moreover, all known Anosov diffeomorphisms are conjugate to the
algebraic Anosov automorphisms.
![Page 49: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/49.jpg)
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
If M supports an Anosov diffeomorphism then M is a torus Rk/Zk ,
a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact
lattice), or an infranilmanifold.
Instead of requiring TxM to have a hyperbolic splitting E sx ⊕ Eu
x
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.
![Page 50: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/50.jpg)
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
If M supports an Anosov diffeomorphism then M is a torus Rk/Zk ,
a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact
lattice), or an infranilmanifold.
Instead of requiring TxM to have a hyperbolic splitting E sx ⊕ Eu
x
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.
![Page 51: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/51.jpg)
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
If M supports an Anosov diffeomorphism then M is a torus Rk/Zk ,
a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact
lattice), or an infranilmanifold.
Instead of requiring TxM to have a hyperbolic splitting E sx ⊕ Eu
x
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.
![Page 52: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/52.jpg)
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
If M supports an Anosov diffeomorphism then M is a torus Rk/Zk ,
a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact
lattice), or an infranilmanifold.
Instead of requiring TxM to have a hyperbolic splitting E sx ⊕ Eu
x
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.
![Page 53: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/53.jpg)
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
If M supports an Anosov diffeomorphism then M is a torus Rk/Zk ,
a nilmanifold (N/Γ where N is a nilpotent Lie group, Γ a compact
lattice), or an infranilmanifold.
Instead of requiring TxM to have a hyperbolic splitting E sx ⊕ Eu
x
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.
![Page 54: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/54.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 55: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/55.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 56: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/56.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 57: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/57.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 58: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/58.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 59: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/59.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 60: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/60.jpg)
DefinitionA compact T -invariant set Λ ⊂ M is a locally maximal hyperbolic
set (or basic set) if
1. there exists a splitting TxM = E sx ⊕ Eu
x such that for all x ∈ Λ
‖DxTn(v)‖ ≤ Cλn‖v‖ ∀v ∈ E s
x , ∀n ≥ 0
‖DxT−n(v)‖ ≤ Cλn‖v‖ ∀v ∈ Eu
x , ∀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.
∞⋂n=−∞
T nU = Λ.
If in (3) we have⋂∞
n=0 T nU = Λ, then Λ is an attractor.
Anosov diffeomorphisms are attractors (with Λ = U = M).
![Page 61: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/61.jpg)
Example 1: The Smale horseshoe
Define
T
If one iterates T forwards:
![Page 62: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/62.jpg)
Example 1: The Smale horseshoe
Define
T
If one iterates T forwards:
![Page 63: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/63.jpg)
Example 1: The Smale horseshoe
Define
T
If one iterates T forwards:
![Page 64: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/64.jpg)
Example 1: The Smale horseshoe
Define
T
If one iterates T forwards:
![Page 65: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/65.jpg)
and backwards
T−1
There is a locally maximal hyperbolic set
-the product of two Cantor sets.
![Page 66: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/66.jpg)
and backwards
T−1
There is a locally maximal hyperbolic set
-the product of two Cantor sets.
![Page 67: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/67.jpg)
and backwards
T−1
There is a locally maximal hyperbolic set
-the product of two Cantor sets.
![Page 68: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/68.jpg)
and backwards
T−1
There is a locally maximal hyperbolic set
-the product of two Cantor sets.
![Page 69: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/69.jpg)
and backwards
T−1
There is a locally maximal hyperbolic set
-the product of two Cantor sets.
![Page 70: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/70.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 71: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/71.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus.
Define a map T : M → M
Take a cross-section of the torus:
![Page 72: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/72.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 73: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/73.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 74: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/74.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 75: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/75.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 76: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/76.jpg)
Example 2: The Solenoid Attractor
Let M = solid torus. Define a map T : M → M
Take a cross-section of the torus:
![Page 77: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/77.jpg)
So Λ =⋂∞
n=0 T nM 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?)
![Page 78: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/78.jpg)
So Λ =⋂∞
n=0 T nM 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?)
![Page 79: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/79.jpg)
So Λ =⋂∞
n=0 T nM 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?)
![Page 80: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/80.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 81: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/81.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 82: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/82.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 83: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/83.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 84: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/84.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 85: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/85.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 86: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/86.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 87: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/87.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 88: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/88.jpg)
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
W sε (x) = {y ∈ M | d(T nx ,T ny) ≤ ε, ∀n ≥ 0}
W uε (x) = {y ∈ M | d(T−nx ,T−ny) ≤ ε, ∀n ≥ 0}.
(It follows that d(T nx ,T ny)→ 0 exponentially fast as n→ ±∞respectively.)
![Page 89: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/89.jpg)
If x , y ∈ Λ are sufficiently close then we define their “product” to
be
[x , y ] = W uε (x) ∩W s
ε (y).
Thus [x , y ] is a point whose orbit approximates that of y (in
forward time) and approximates x (in backwards time).
![Page 90: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/90.jpg)
If x , y ∈ Λ are sufficiently close then we define their “product” to
be
[x , y ] = W uε (x) ∩W s
ε (y).
Thus [x , y ] is a point whose orbit approximates that of y (in
forward time) and approximates x (in backwards time).
![Page 91: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/91.jpg)
If x , y ∈ Λ are sufficiently close then we define their “product” to
be
[x , y ] = W uε (x) ∩W s
ε (y).
Thus [x , y ] is a point whose orbit approximates that of y (in
forward time) and approximates x (in backwards time).
![Page 92: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/92.jpg)
Example: The Cat Map
y
xW uε (y)
![Page 93: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/93.jpg)
Example: The Cat Map
y
xW uε (x)
![Page 94: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/94.jpg)
Example: The Cat Map
y
xW uε (x)
![Page 95: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/95.jpg)
Example: The Cat Map
y
W sε (y)
xW uε (x)
![Page 96: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/96.jpg)
Example: The Cat Map
[x , y ]
y
W sε (y)
xW uε (x)
![Page 97: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/97.jpg)
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
W s(x ,R) = W sε ∩ R
W u(x ,R) = W uε ∩ R.
x
W s(x ,R)
W u(x ,R)
![Page 98: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/98.jpg)
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
W s(x ,R) = W sε ∩ R
W u(x ,R) = W uε ∩ R.
x
W s(x ,R)
W u(x ,R)
![Page 99: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/99.jpg)
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
W s(x ,R) = W sε ∩ R
W u(x ,R) = W uε ∩ R.
x
W s(x ,R)
W u(x ,R)
![Page 100: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/100.jpg)
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
W s(x ,R) = W sε ∩ R
W u(x ,R) = W uε ∩ R.
x
W s(x ,R)
W u(x ,R)
![Page 101: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/101.jpg)
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
W s(x ,R) = W sε ∩ R
W u(x ,R) = W uε ∩ R.
x
W s(x ,R)
W u(x ,R)
![Page 102: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/102.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 103: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/103.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R
=⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 104: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/104.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 105: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/105.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 106: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/106.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 107: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/107.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
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A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 109: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/109.jpg)
A partition R = {R1, . . . ,Rk} of Λ into rectangles is called a
Markov partition if
x ∈ Int R =⇒
T (W u(x ,R)) is a union of sets of the
form W u(y ,R ′).
T−1(W s(y ,R)) is a union of sets of
the form W s(y ,R ′).
x
W u(x ,R)
R1
R2
R3
R4
TW u(x ,R)
![Page 110: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/110.jpg)
Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4.
For
x ∈ ∩∞n=−∞T n(IntR), we can code the orbit of x by recording the
sequence of rectangles the orbit visits. Note that (cf decimal
expansions) if x ∈⋃∞
n=−∞ T n(∂R) then the coding is not unique.
![Page 111: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/111.jpg)
Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4. For
x ∈ ∩∞n=−∞T n(IntR), we can code the orbit of x by recording the
sequence of rectangles the orbit visits.
Note that (cf decimal
expansions) if x ∈⋃∞
n=−∞ T n(∂R) then the coding is not unique.
![Page 112: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/112.jpg)
Idea: If x ∈ R1, then T (x) must be in either R2, R3, R4. For
x ∈ ∩∞n=−∞T n(IntR), we can code the orbit of x by recording the
sequence of rectangles the orbit visits. Note that (cf decimal
expansions) if x ∈⋃∞
n=−∞ T n(∂R) then the coding is not unique.
![Page 113: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/113.jpg)
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
π : Σ −→ Λ : (xj)∞−∞ 7−→
∞⋂−∞
T−jRxj
is continuous, surjective, and injective except on⋃∞−∞ T−j(∂R)
2.
Σσ //
π
��
Σ
π
��Λ
T// Λ
commutes.
![Page 114: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/114.jpg)
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
π : Σ −→ Λ : (xj)∞−∞ 7−→
∞⋂−∞
T−jRxj
is continuous, surjective, and injective except on⋃∞−∞ T−j(∂R)
2.
Σσ //
π
��
Σ
π
��Λ
T// Λ
commutes.
![Page 115: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/115.jpg)
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
π : Σ −→ Λ : (xj)∞−∞ 7−→
∞⋂−∞
T−jRxj
is continuous, surjective, and injective except on⋃∞−∞ T−j(∂R)
2.
Σσ //
π
��
Σ
π
��Λ
T// Λ
commutes.
![Page 116: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/116.jpg)
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
π : Σ −→ Λ : (xj)∞−∞ 7−→
∞⋂−∞
T−jRxj
is continuous, surjective, and injective except on⋃∞−∞ T−j(∂R)
2.
Σσ //
π
��
Σ
π
��Λ
T// Λ
commutes.
![Page 117: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/117.jpg)
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
π : Σ −→ Λ : (xj)∞−∞ 7−→
∞⋂−∞
T−jRxj
is continuous, surjective, and injective except on⋃∞−∞ T−j(∂R)
2.
Σσ //
π
��
Σ
π
��Λ
T// Λ
commutes.
![Page 118: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/118.jpg)
Example: Cat Map
This gives the matrix:0BBBB@1 1 0 1 0
1 1 0 1 0
1 1 0 1 0
0 0 1 0 1
0 0 1 0 1
1CCCCA
![Page 119: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/119.jpg)
Example: Cat Map
This gives the matrix:0BBBB@1 1 0 1 0
1 1 0 1 0
1 1 0 1 0
0 0 1 0 1
0 0 1 0 1
1CCCCA
![Page 120: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/120.jpg)
Example: Cat Map
This gives the matrix:0BBBB@1 1 0 1 0
1 1 0 1 0
1 1 0 1 0
0 0 1 0 1
0 0 1 0 1
1CCCCA
![Page 121: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/121.jpg)
Example: Cat Map
R0
R3
R4
R1
R2
0BBBB@1 1 0 1 0
1 1 0 1 0
1 1 0 1 0
0 0 1 0 1
0 0 1 0 1
1CCCCA
This gives the matrix:
![Page 122: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/122.jpg)
Example: Cat Map
R0
R3
R4
R1
R2
0BBBB@1 1 0 1 0
1 1 0 1 0
1 1 0 1 0
0 0 1 0 1
0 0 1 0 1
1CCCCA
This gives the matrix:
![Page 123: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/123.jpg)
RemarkIn 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.
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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.
![Page 125: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/125.jpg)
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.
![Page 126: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/126.jpg)
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.
![Page 127: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/127.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 128: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/128.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 129: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/129.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 130: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/130.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 131: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/131.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 132: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/132.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 133: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/133.jpg)
Functions of the future
Let Σ = two sided shift of finite type
dθ = metric given by θ|first disagreement|
Fθ(Σ,R) = {functions f : Σ→ R that are θ-Holder}.
If x = (xj)∞j=−∞ ∈ Σ then we think of (xj)
∞j=0 as “the future” and
(xj)0−∞ as “the past”.
Note that if f ∈ Fθ(Σ,R) then, typically, f (x) will depend both on
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.
![Page 134: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/134.jpg)
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∑j=0
f (σjx) =n−1∑j=0
g(σjx) + u(σnx)− u(x)
=n−1∑j=0
g(σjx) + O(1).
TheoremLet f ∈ Fθ(Σ,R). Then f is cohomologous to a function
g ∈ Fθ
12
(Σ+,R) that depends only on the future.
![Page 135: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/135.jpg)
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∑j=0
f (σjx) =n−1∑j=0
g(σjx) + u(σnx)− u(x)
=n−1∑j=0
g(σjx) + O(1).
TheoremLet f ∈ Fθ(Σ,R). Then f is cohomologous to a function
g ∈ Fθ
12
(Σ+,R) that depends only on the future.
![Page 136: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/136.jpg)
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∑j=0
f (σjx) =n−1∑j=0
g(σjx) + u(σnx)− u(x)
=n−1∑j=0
g(σjx) + O(1).
TheoremLet f ∈ Fθ(Σ,R). Then f is cohomologous to a function
g ∈ Fθ
12
(Σ+,R) that depends only on the future.
![Page 137: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/137.jpg)
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∑j=0
f (σjx) =n−1∑j=0
g(σjx) + u(σnx)− u(x)
=n−1∑j=0
g(σjx) + O(1).
TheoremLet f ∈ Fθ(Σ,R). Then f is cohomologous to a function
g ∈ Fθ
12
(Σ+,R) that depends only on the future.
![Page 138: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/138.jpg)
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
3. if f : Λ→ R is Holder then f = f π ∈ Fθ(Σ,R) for some
θ ∈ (0, 1).
4. replace f by a cohomologous function g ∈ Fθ
12
(Σ+,R) and
apply thermodynamic formalism.
![Page 139: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/139.jpg)
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
3. if f : Λ→ R is Holder then f = f π ∈ Fθ(Σ,R) for some
θ ∈ (0, 1).
4. replace f by a cohomologous function g ∈ Fθ
12
(Σ+,R) and
apply thermodynamic formalism.
![Page 140: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/140.jpg)
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
3. if f : Λ→ R is Holder then f = f π ∈ Fθ(Σ,R) for some
θ ∈ (0, 1).
4. replace f by a cohomologous function g ∈ Fθ
12
(Σ+,R) and
apply thermodynamic formalism.
![Page 141: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/141.jpg)
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
3. if f : Λ→ R is Holder then f = f π ∈ Fθ(Σ,R) for some
θ ∈ (0, 1).
4. replace f by a cohomologous function g ∈ Fθ
12
(Σ+,R) and
apply thermodynamic formalism.
![Page 142: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/142.jpg)
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
3. if f : Λ→ R is Holder then f = f π ∈ Fθ(Σ,R) for some
θ ∈ (0, 1).
4. replace f by a cohomologous function g ∈ Fθ
12
(Σ+,R) and
apply thermodynamic formalism.
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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 Holder:
|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.
Let µf = νf ◦ π−1. Then µf is called an equilibrium state for f and
is a T -invariant measure.
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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 Holder:
|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.
Let µf = νf ◦ π−1. Then µf is called an equilibrium state for f and
is a T -invariant measure.
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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 Holder:
|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.
Let µf = νf ◦ π−1. Then µf is called an equilibrium state for f and
is a T -invariant measure.
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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 Holder:
|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.
Let µf = νf ◦ π−1. Then µf is called an equilibrium state for f and
is a T -invariant measure.
![Page 147: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/147.jpg)
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
limn→∞
1
n
n−1∑j=0
f (T jx)
for m-a.e. x ∈ X?
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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
limn→∞
1
n
n−1∑j=0
f (T jx)
for m-a.e. x ∈ X?
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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.
1
n
n−1∑j=0
f (T jx) −→∫
f dµ ∀x ∈ N.
(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).
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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.
1
n
n−1∑j=0
f (T jx) −→∫
f dµ ∀x ∈ N.
(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).
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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.
1
n
n−1∑j=0
f (T jx) −→∫
f dµ ∀x ∈ N.
(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).
![Page 152: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/152.jpg)
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.
1
n
n−1∑j=0
f (T jx) −→∫
f dµ ∀x ∈ N.
(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).
![Page 153: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/153.jpg)
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.
1
n
n−1∑j=0
f (T jx) −→∫
f dµ ∀x ∈ N.
(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).
![Page 154: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/154.jpg)
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?
i.e. does limn→∞1n
∑n−1j=0 f (T jx) exist for all con-
tinuous f , m-a.e., and what is the limit?
![Page 155: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/155.jpg)
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?
i.e. does limn→∞1n
∑n−1j=0 f (T jx) exist for all con-
tinuous f , m-a.e., and what is the limit?
![Page 156: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/156.jpg)
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?
i.e. does limn→∞1n
∑n−1j=0 f (T jx) exist for all con-
tinuous f , m-a.e., and what is the limit?
![Page 157: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/157.jpg)
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?
i.e. does limn→∞1n
∑n−1j=0 f (T jx) exist for all con-
tinuous f , m-a.e., and what is the limit?
![Page 158: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/158.jpg)
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?
i.e. does limn→∞1n
∑n−1j=0 f (T jx) exist for all con-
tinuous f , m-a.e., and what is the limit?
![Page 159: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/159.jpg)
Example: The North-South Map T : S1 → S1
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
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Example: The North-South Map T : S1 → S1
r
rS
N
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
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Example: The North-South Map T : S1 → S1
r
r
r
S
x
N
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 162: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/162.jpg)
Example: The North-South Map T : S1 → S1
~
r
r
r
S
x
N
u
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 163: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/163.jpg)
Example: The North-South Map T : S1 → S1
~
r
r
r
S
x
N
u2
u
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 164: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/164.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r
r
S
x
N
u2
u
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 165: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/165.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r rr
S
Tx
x
N
u2
u
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 166: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/166.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r rr
S
Tx
x
N
u2
u
T (N) = N
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 167: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/167.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r rr
S
Tx
x
N
u2
u
T (N) = N
Then S is an attractor with basin S1 \ {N}.
Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 168: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/168.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r rr
S
Tx
x
N
u2
u
T (N) = N
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R).
As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 169: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/169.jpg)
Example: The North-South Map T : S1 → S1
K~
r
r rr
S
Tx
x
N
u2
u
T (N) = N
Then S is an attractor with basin S1 \ {N}. Let f ∈ C (X ,R). As
T nx → S ∀x ∈ S1 \ {N}, we have
1
n
n−1∑j=0
f (T jx) −→ f (S) =
∫f dδS .
![Page 170: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/170.jpg)
DefinitionLet T : Λ→ Λ be an attractor. A T -invariant probability measure
µ is an SRB (Sinai-Ruelle-Bowen) measure if
1
n
n−1∑j=0
f (T jx) −→∫
f dµ
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.
![Page 171: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/171.jpg)
DefinitionLet T : Λ→ Λ be an attractor. A T -invariant probability measure
µ is an SRB (Sinai-Ruelle-Bowen) measure if
1
n
n−1∑j=0
f (T jx) −→∫
f dµ
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.
![Page 172: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/172.jpg)
DefinitionLet T : Λ→ Λ be an attractor. A T -invariant probability measure
µ is an SRB (Sinai-Ruelle-Bowen) measure if
1
n
n−1∑j=0
f (T jx) −→∫
f dµ
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.
![Page 173: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/173.jpg)
DefinitionLet T : Λ→ Λ be an attractor. A T -invariant probability measure
µ is an SRB (Sinai-Ruelle-Bowen) measure if
1
n
n−1∑j=0
f (T jx) −→∫
f dµ
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.
![Page 174: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/174.jpg)
TheoremLet 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 |Eu
RemarkSuppose 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.
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TheoremLet 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 |Eu
RemarkSuppose 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.
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TheoremLet 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 |Eu
RemarkSuppose 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.
![Page 177: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/177.jpg)
TheoremLet 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 |Eu
RemarkSuppose 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.
![Page 178: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/178.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 179: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/179.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 180: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/180.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 181: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/181.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 182: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/182.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 183: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/183.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...
![Page 184: MAGIC: Ergodic Theory Lecture 10 - The ergodic theory of ... · ergodic theory of hyperbolic dynamical systems Charles Walkden April 12, 2013. In the last two lectures we studied](https://reader034.vdocument.in/reader034/viewer/2022050104/5f42ea1d5323535ba4389ecc/html5/thumbnails/184.jpg)
(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 ⊕ Eu,
I thermodynamic formalism for countable state shifts of finite
type,
I ...