The Structure of
Hyperbolic Sets
Todd Fisher
[email protected]
Department of Mathematics
University of Maryland, College Park
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Outline of Talk
History and Examples
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Outline of Talk
History and Examples
Properties
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Outline of Talk
History and Examples
Properties
Structure
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3 ways dynamics
studied
Measurable:
Probabilistic and statistical
properties.
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3 ways dynamics
studied
Measurable:
Probabilistic and statistical
properties.
This studies functions that are
only assumed to be continuous.
Topological:
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3 ways dynamics
studied
Measurable:
Probabilistic and statistical
properties.
This studies functions that are
only assumed to be continuous.
Topological:
Smooth:
Assume there is a derivative at every
point.
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Advantages to
Smooth
The local picture given by derivative
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Advantages to
Smooth
The local picture given by derivative
Very useful in hyperbolic case. Tangent space
TΛ M splits into expanding E u and contracting
directions E s.
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Advantages to
Smooth
The local picture given by derivative
Very useful in hyperbolic case. Tangent space
TΛ M splits into expanding E u and contracting
directions E s.
For instance, say
"
#" #
1/2 0 x
f (x, y) =
0 2 y
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Stable set and
Unstable set
The stable set of a point x ∈ M is
W s (x) = {y ∈ M |d(f n (x), f n (y)) → 0 as n → ∞}.
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Stable set and
Unstable set
The stable set of a point x ∈ M is
W s (x) = {y ∈ M |d(f n (x), f n (y)) → 0 as n → ∞}.
The unstable set of a point x ∈ M is
W u (x) = {y ∈ M |d(f −n (x), f −n (y)) → 0
as n
→ ∞}.
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One Origin - Celestial
Mechanics
In 19th century Poincaré began to study stability
of solar system.
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One Origin - Celestial
Mechanics
In 19th century Poincaré began to study stability
of solar system.
For a flow from a differential equation with fixed
hyperbolic saddle point p and point
x ∈ W s (p) ∩ W u (p).
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One Origin - Celestial
Mechanics
In 19th century Poincaré began to study stability
of solar system.
For a flow from a differential equation with fixed
hyperbolic saddle point p and point
x ∈ W s (p) ∩ W u (p).
P
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Transverse
Homoclinic Point
If we look at a function f picture can become
more complicated. This was in some sense the
start of chaotic dynamics.
A point x ∈ W s (p) ⋔ W u (p) is called a transverse
homoclinic point.
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Homoclinic Tangle
x
p
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Homoclinic Tangle
x
f(x)
p
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Homoclinic Tangle
x
f(x)
p
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Homoclinic Tangle
x
f(x)
p
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Further Results on
Homoclinic Points
In 1930’s Birkhoff showed that near a transverse
homoclinic point ∃ pn → x such that pn periodic
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Further Results on
Homoclinic Points
In 1930’s Birkhoff showed that near a transverse
homoclinic point ∃ pn → x such that pn periodic
In 1960’s Smale showed the following:
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Further Results on
Homoclinic Points
In 1930’s Birkhoff showed that near a transverse
homoclinic point ∃ pn → x such that pn periodic
In 1960’s Smale showed the following:
n
f (D)
D
p
x
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Smale’s Horseshoe
Smale generalized picture as follows:
Take the unit square R = [0, 1] × [0, 1] map the
square as shown below.
f(R)
A
1111111
0000000
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1111111
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1111111
000000
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1111111
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1111111
000000
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0000000
1111111
000000
111111
0000000
1111111
000000
111111
000000
R 111111
000000
111111
000000
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000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
11111111
00000000
000000
111111
00000000
11111111
000000
111111
00000000
11111111
000000
111111
00000000
11111111
000000
111111
00000000
11111111
000000
111111
00000000
11111111
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
000000
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000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
B
There exist two region"A and #
B in R such that
1/3 0
f |A and f |B looks like
0 3
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Invariant Set for
Horseshoe - 1
We want points that never leave R.
\
Λ=
f n (R)
n∈Z
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Invariant Set for
Horseshoe -3
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Invariant Set for
Horseshoe -3
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Invariant Set for
Horseshoe -3
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Dynamics of
Horseshoe
Then Λ is Middle Thirds Cantor × Middle Thirds
Cantor
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Dynamics of
Horseshoe
Then Λ is Middle Thirds Cantor × Middle Thirds
Cantor
The set Λ is chaotic in the sense of Devaney.
periodic points of Λ are dense
there is a point with a dense orbit (transitive)
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Dynamics of
Horseshoe
Then Λ is Middle Thirds Cantor × Middle Thirds
Cantor
The set Λ is chaotic in the sense of Devaney.
periodic points of Λ are dense
there is a point with a dense orbit (transitive)
So Horseshoe is very interesting dynamically.
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Hyperbolic Set
A compact set Λ is hyperbolic if it is
invariant (f (Λ) = Λ) and the tangent space has a
continuous invariant splitting TΛ M = Es ⊕ Eu
where Es is uniformly contracting and Eu is
uniformly expanding.
Hyperbolic
So ∃ C > 0 and λ ∈ (0, 1) such that:
kDfxn vk ≤ Cλn kvk ∀ v ∈ Esx , n ∈ N and
kDfx−n vk ≤ Cλn kvk ∀ v ∈ Eux , n ∈ N
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Hyperbolic
Properties -1
For a point of a hyperbolic set x ∈ Λ the stable
and unstable sets are immersed copies of Rm
and Rn where m = dim(E s ) and n = dim(E u ).
Tx W s (x) = Exs and Tx W u (x) = Exu
Closed + Bounded + Hyperbolic = Interesting
Dynamics
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Morse-Smale
Diffeormophisms
A diffeo. f is Morse-Smale if the only recurrent
points are a finite number hyperbolic periodic
points and the stable and unstable manifolds of
each periodic point is transverse.
so dynamics are gradient like.
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Weak Palis
Conjecture
Recent Theorem says horseshoes are very
common for diffeomorphisms.
Note:
Theorem 1 (Weak Palis Conjecture) For any
smooth manifold the set of there is an open and
dense set of C 1 diffeomorphisms that are either
Morse-Smale or contain a horseshoe.
Proof announced by Crovisier, based on work of
Bonatti, Gan, and Wen.
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Hyperbolic Toral
Automorphisms
"
#
2 1
Take the Matrix A =
.
1 1
This matrix has det(A) = 1, one eigenvalue
λs ∈ (0, 1), and one eigenvalue λu ∈ (1, ∞). So
one contracting direction and one expanding
direction.
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Anosov
Diffeomorphisms
Since A has determinant 1 it preserves Z2 there
is induced map on torus fA from A. At every
point x ∈ T2 there is a contacting and expanding
direction.
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Anosov
Diffeomorphisms
Since A has determinant 1 it preserves Z2 there
is induced map on torus fA from A. At every
point x ∈ T2 there is a contacting and expanding
direction.
A diffeomorphism is Anosov if the entire manifold
is a hyperbolic set. So fA is Anosov.
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Anosov Diffeos in
2-dimensions
In two dimensions only the torus supports
Anosov diffeomorphisms and all are topologically
conjugate to hyperbolic toral automorphisms.
Two maps f : X → X and g : Y → Y are
conjugate if there is a continuous
homeomorphism h : X → Y such that hf = gh.
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Attractors
Definition 2 A set X has an attracting
neighborhood
if ∃ neighborhood U of X such that
T
X = n∈N f n (U ).
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Attractors
Definition 3 A set X has an attracting
neighborhood
if ∃ neighborhood U of X such that
T
X = n∈N f n (U ).
A hyperbolic set Λ is a hyperbolic attractor if Λ is
transitive (contains a point with a dense orbit)
and has an attracting neighborhood.
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Attractors
Definition 4 A set X has an attracting
neighborhood
if ∃ neighborhood U of X such that
T
X = n∈N f n (U ).
A hyperbolic set Λ is a hyperbolic attractor if Λ is
transitive (contains a point with a dense orbit)
and has an attracting neighborhood.
For a compact surface result of Plykin says there
must be at least 3 holes for a hyperbolic attractor.
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Plykin Attractor
V
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Plykin Attractor
V
f(V)
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Dynamics of
Attractors
V is an attracting neighborhood and
\
Λ=
f n (V ).
n∈N
In two dimensions a hyperbolic attractor looks
locally like a Cantor set × interval.
The interval is the unstable direction the Cantor
set is the stable direction.
Hyperbolic attractors have dense periodic points
and a point with a dense orbit.
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Locally Maximal
A hyperbolic set Λ is locally maximal (or isolated) if
∃ open set U such that
\
Λ=
f n (U )
n∈Z
All the examples we looked at are locally
maximal
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Properties of Locally
Maximal Sets
Locally maximal transitive hyperbolic sets have
nice properties including:
1. Shadowing
2. Structural Stability
3. Markov Partitions
4. SRB measures (for attractors)
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Shadowing
A sequence x1 , x2 , ..., xn is an ǫ pseudo-orbit if
d(f (xi ), xi+1 ) < ǫ for all 1 ≤ i < n.
A point y δ -shadows an ǫ pseudo-orbit if
d(f i (y), xi ) < δ for all 1 ≤ i ≤ n.
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Shadowing Diagram
f(x1)
x1
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Shadowing Diagram
f(x1)
x2
x1
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Shadowing Diagram
f(x1)
x2
x3
f(x2)
x1
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Shadowing Diagram
f(x1)
y
x2
x3
f(x2)
x1
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Shadowing Diagram
f(x1)
y
f(y)
x2
x3
f(x2)
x1
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Shadowing Diagram
2
f(x1)
y
f(y)
x2
f (y)
x3
f(x2)
x1
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Shadowing Theorem
Theorem 5 (Shadowing Theorem) Let Λ be a
compact hyperbolic invariant set. Given δ > 0 ∃
j2
ǫ, η > 0 such that if {xj }j=j1 is an ǫ pseudo-orbit
for f with d(xj , Λ) < η for j1 ≤ j ≤ j2 , then ∃ y
which δ-shadows {xj }. If j1 = −∞ and j2 = ∞,
then y is unique. If Λ is locally maximal and
j1 = −∞ and j2 = ∞, then y ∈ Λ.
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Structural Stability
Theorem 6 (Structural Stability) If Λ is a
hyperbolic set for f , then there exists a C 1 open
set U containing f such that for all g ∈ U there
exists a hyperbolic set Λg and homeomorphism
h : Λ → Λg such that hf = gh.
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Markov Partition
Decomposition of Λ into finite number of
dynamical rectangles R1 , ..., Rn such that for
each 1 ≤ i ≤ n
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Markov Partition
Decomposition of Λ into finite number of
dynamical rectangles R1 , ..., Rn such that for
each 1 ≤ i ≤ n
int(Ri ) ∩ int(Rj ) = ∅ if i 6= j
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Markov Partition
Decomposition of Λ into finite number of
dynamical rectangles R1 , ..., Rn such that for
each 1 ≤ i ≤ n
int(Ri ) ∩ int(Rj ) = ∅ if i 6= j
for some ǫ sufficiently small Ri is
(Wǫu (x) ∩ Ri ) × (Wǫs (x) ∩ Ri )
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Markov Partition
Decomposition of Λ into finite number of
dynamical rectangles R1 , ..., Rn such that for
each 1 ≤ i ≤ n
int(Ri ) ∩ int(Rj ) = ∅ if i 6= j
for some ǫ sufficiently small Ri is
(Wǫu (x) ∩ Ri ) × (Wǫs (x) ∩ Ri )
x ∈ Ri , f (x) ∈ Rj , and i → j is an allowed
transition in Σ, then
f (W s (x, Ri )) ⊂ Rj and f −1 (W u (f (x), Rj )) ⊂ Ri .
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SRB Measurs
If Λ is a hyperbolic attractor ∃ measure µ on Λ
such that for a.e x in basin of attraction and any
observable φ:
1 n
lim Σi=1 φ(f i (x)) =
n→∞ n
Z
φ dµ
Λ
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Question 1
Katok and Hasselblatt:
Question 1: “Let Λ be a hyperbolic set... and V an
open neighborhood of Λ. Does there exist a
locally maximal hyperbolic set Λ̃ such that
Λ ⊂ Λ̃ ⊂ V ?”
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Question 1
Katok and Hasselblatt:
Question 1: “Let Λ be a hyperbolic set... and V an
open neighborhood of Λ. Does there exist a
locally maximal hyperbolic set Λ̃ such that
Λ ⊂ Λ̃ ⊂ V ?”
Crovisier(2001) answers no for specific example
on four torus.
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Question 1
Katok and Hasselblatt:
Question 1: “Let Λ be a hyperbolic set... and V an
open neighborhood of Λ. Does there exist a
locally maximal hyperbolic set Λ̃ such that
Λ ⊂ Λ̃ ⊂ V ?”
Crovisier(2001) answers no for specific example
on four torus.
Related questions:
1. Can this be robust?
2. Can this happen on other manifolds, in lower
dimension, on all manifolds?
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Robust and Markov
Theorems
Theorem 7 (F.) On any compact manifold M ,
where dim(M ) ≥ 2, there exists a C 1 open set of
diffeomorphisms, U, such that any f ∈ U has a
hyperbolic set that is not contained in a locally
maximal hyperbolic set.
Theorem 8 (F.) If Λ is a hyperbolic set and V is
a neighborhood of Λ, then there exists a
hyperbolic set Λ̃ with a Markov partition such that
Λ ⊂ Λ̃ ⊂ V .
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Hyperbolic Sets with
Interior
Theorem 9 (F.) If Λ is a hyperbolic set with
nonempty interior, then f is Anosov if
1. Λ is transitive
2. Λ is locally maximal and M is a surface
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Hyperbolic Attractors
on Surfaces
Theorem 10 (F.) If M is a compact smooth
surface, Λ is a hyperbolic attractor for f , and
hyperbolic for g, then Λ is either a hyperbolic
attractor or repeller for g.
So if we know the topology of the set and we
know that it is hyperbolic we know it is an
attractor.
A set Λ is a repeller
exists neighborhood
T if there
V such that Λ = n∈N f −n (V ).
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