The Structure of Hyperbolic Setstfisher/documents/presentations/hypsets.pdfThe Structure of...

The Structure ofHyperbolic Sets

Todd Fisher

[email protected]

Department of Mathematics

University of Maryland, College Park

The Structure of Hyperbolic Sets – p. 1/35

Outline of TalkHistory and Examples



Properties



Properties

Structure


3 ways dynamicsstudied

Measurable: Probabilistic and statisticalproperties.




Topological: This studies functions that areonly assumed to be continuous.




Topological: This studies functions that areonly assumed to be continuous.

Smooth: Assume there is a derivative at everypoint.


Advantages toSmooth

The local picture given by derivative


Advantages toSmooth


Very useful in hyperbolic case. Tangent spaceTΛM splits into expanding Eu and contractingdirections Es.


Advantages toSmooth


Very useful in hyperbolic case. Tangent spaceTΛM splits into expanding Eu and contractingdirections Es.

For instance, say

f(x, y) =

[

1/2 0

0 2

] [

x

y

]


Stable set andUnstable set

The stable set of a point x ∈ M is

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


Stable set andUnstable set

The stable set of a point x ∈ M is

W s(x) = {y ∈ M |d(fn(x), fn(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 → ∞}.


One Origin - CelestialMechanics

In 19th century Poincaré began to study stabilityof solar system.




For a flow from a differential equation with fixedhyperbolic saddle point p and pointx ∈ W s(p) ∩ W u(p).




For a flow from a differential equation with fixedhyperbolic saddle point p and pointx ∈ W s(p) ∩ W u(p).

P


TransverseHomoclinic Point

If we look at a function f picture can becomemore complicated. This was in some sense thestart of chaotic dynamics.

A point x ∈ W s(p) ⋔ W u(p) is called a transversehomoclinic point.


Homoclinic Tangle

p

x


Homoclinic Tangle

f(x)

x

p


Homoclinic Tangle

p

x

f(x)


Further Results onHomoclinic Points

In 1930’s Birkhoff showed that near a transversehomoclinic point ∃ pn → x such that pn periodic




In 1960’s Smale showed the following:




In 1960’s Smale showed the following:

nf (D)

D

p x


Smale’s HorseshoeSmale generalized picture as follows:Take the unit square R = [0, 1] × [0, 1] map thesquare as shown below.

BA

f(R)

R��

��

��

��

��

��

��

��

There exist two region A and B in R such that

f |A and f |B looks like

[

1/3 0

0 3

]


Invariant Set forHorseshoe - 1

We want points that never leave R.

Λ =⋂

n∈Z

fn(R)


Invariant Set forHorseshoe -3


Dynamics ofHorseshoe

Then Λ is Middle Thirds Cantor × Middle ThirdsCantor




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)




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.


Hyperbolic Set

Hyperbolic A compact set Λ is hyperbolic if it isinvariant (f(Λ) = Λ) and the tangent space has acontinuous invariant splitting TΛM = E

s ⊕ Eu

where Es is uniformly contracting and E

u isuniformly expanding.

So ∃ C > 0 and λ ∈ (0, 1) such that:‖Dfn

x v‖ ≤ Cλn‖v‖ ∀ v ∈ Esx, n ∈ N and

‖Df−nx v‖ ≤ Cλn‖v‖ ∀ v ∈ E

ux, n ∈ N


HyperbolicProperties -1

For a point of a hyperbolic set x ∈ Λ the stableand unstable sets are immersed copies of R

m

and Rn where m = dim(Es) and n = dim(Eu).

TxWs(x) = Es

x and TxWu(x) = Eu

x

Closed + Bounded + Hyperbolic = InterestingDynamics


Morse-SmaleDiffeormophisms

A diffeo. f is Morse-Smale if the only recurrentpoints are a finite number hyperbolic periodicpoints and the stable and unstable manifolds ofeach periodic point is transverse.

so dynamics are gradient like.


Weak PalisConjecture

Note: Recent Theorem says horseshoes are verycommon for diffeomorphisms.

Theorem 1 (Weak Palis Conjecture) For anysmooth manifold the set of there is an open anddense set of C1 diffeomorphisms that are eitherMorse-Smale or contain a horseshoe.

Proof announced by Crovisier, based on work ofBonatti, Gan, and Wen.


Hyperbolic ToralAutomorphisms

Take the Matrix A =

[

2 1

1 1

]

.

This matrix has det(A) = 1, one eigenvalueλs ∈ (0, 1), and one eigenvalue λu ∈ (1,∞). Soone contracting direction and one expandingdirection.


AnosovDiffeomorphisms

Since A has determinant 1 it preserves Z2 there

is induced map on torus fA from A. At everypoint x ∈ T

2 there is a contacting and expandingdirection.


AnosovDiffeomorphisms

Since A has determinant 1 it preserves Z2 there

is induced map on torus fA from A. At everypoint x ∈ T

2 there is a contacting and expandingdirection.

A diffeomorphism is Anosov if the entire manifoldis a hyperbolic set. So fA is Anosov.


Anosov Diffeos in2-dimensions

In two dimensions only the torus supportsAnosov diffeomorphisms and all are topologicallyconjugate to hyperbolic toral automorphisms.

Two maps f : X → X and g : Y → Y areconjugate if there is a continuoushomeomorphism h : X → Y such that hf = gh.


AttractorsDefinition 2 A set X has an attractingneighborhood if ∃ neighborhood U of X such thatX =

⋂

n∈Nfn(U).



⋂

n∈Nfn(U).

A hyperbolic set Λ is a hyperbolic attractor if Λ istransitive (contains a point with a dense orbit)and has an attracting neighborhood.



⋂

n∈Nfn(U).

A hyperbolic set Λ is a hyperbolic attractor if Λ istransitive (contains a point with a dense orbit)and has an attracting neighborhood.

For a compact surface result of Plykin says theremust be at least 3 holes for a hyperbolic attractor.


Plykin AttractorV


Plykin AttractorV

f(V)


Dynamics ofAttractors

V is an attracting neighborhood and

Λ =⋂

n∈N

fn(V ).

In two dimensions a hyperbolic attractor lookslocally like a Cantor set × interval.

The interval is the unstable direction the Cantorset is the stable direction.

Hyperbolic attractors have dense periodic pointsand a point with a dense orbit.


Locally Maximal

A hyperbolic set Λ is locally maximal (or isolated ) if∃ open set U such that

Λ =⋂

n∈Z

fn(U)

All the examples we looked at are locallymaximal


Properties of LocallyMaximal Sets

Locally maximal transitive hyperbolic sets havenice properties including:

1. Shadowing

2. Structural Stability

3. Markov Partitions

4. SRB measures (for attractors)


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 ifd(f i(y), xi) < δ for all 1 ≤ i ≤ n.


Shadowing Diagram

1

x1

f(x )


Shadowing Diagram

x

1

2

x1

f(x )


Shadowing Diagram

3

2f(x )

xf(x )

1x

2

1

x


Shadowing Diagram

y3

2f(x )

xf(x )

1x

2

1

x


Shadowing Diagram

f(y)y3

2f(x )

xf(x )

1x

2

1

x


Shadowing Diagram

2f (y)

f(y)y3

2f(x )

xf(x )

1x

2

1

x


Shadowing Theorem

Theorem 5 (Shadowing Theorem) Let Λ be acompact hyperbolic invariant set. Given δ > 0 ∃

ǫ, η > 0 such that if {xj}j2j=j1

is an ǫ pseudo-orbitfor f with d(xj,Λ) < η for j1 ≤ j ≤ j2, then ∃ ywhich δ-shadows {xj}. If j1 = −∞ and j2 = ∞,then y is unique. If Λ is locally maximal andj1 = −∞ and j2 = ∞, then y ∈ Λ.


Structural Stability

Theorem 6 (Structural Stability) If Λ is ahyperbolic set for f , then there exists a C1 openset U containing f such that for all g ∈ U thereexists a hyperbolic set Λg and homeomorphismh : Λ → Λg such that hf = gh.


Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ n


Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j



for some ǫ sufficiently small Ri is(W u

ǫ (x) ∩ Ri) × (W sǫ (x) ∩ Ri)



for some ǫ sufficiently small Ri is(W u

ǫ (x) ∩ Ri) × (W sǫ (x) ∩ Ri)

x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition in Σ, then

f(W s(x,Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.


SRB MeasursIf Λ is a hyperbolic attractor ∃ measure µ on Λsuch that for a.e x in basin of attraction and anyobservable φ:

limn→∞

1

nΣn

i=1φ(f i(x)) =

∫

Λ

φ dµ


Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”


Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”Crovisier (2001) answers no for specific exampleon four torus.


Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”Crovisier (2001) answers no for specific exampleon four torus.Related questions:

1. Can this be robust?

2. Can this happen on other manifolds, in lowerdimension, on all manifolds?


Robust and MarkovTheorems

Theorem 7 (F.) On any compact manifold M ,where dim(M) ≥ 2, there exists a C1 open set ofdiffeomorphisms, U , such that any f ∈ U has ahyperbolic set that is not contained in a locallymaximal hyperbolic set.

Theorem 8 (F.) If Λ is a hyperbolic set and V isa neighborhood of Λ, then there exists ahyperbolic set Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .


Hyperbolic Sets withInterior

Theorem 9 (F.) If Λ is a hyperbolic set withnonempty interior, then f is Anosov if

1. Λ is transitive

2. Λ is locally maximal and M is a surface


Hyperbolic Attractorson Surfaces

Theorem 10 (F.) If M is a compact smoothsurface, Λ is a hyperbolic attractor for f , andhyperbolic for g, then Λ is either a hyperbolicattractor or repeller for g.

So if we know the topology of the set and weknow that it is hyperbolic we know it is anattractor.A set Λ is a repeller if there exists neighborhoodV such that Λ =

⋂

n∈Nf−n(V ).


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