The structure of hyperbolic attractors on surfacestfisher/documents/presentations/...The structure of hyperbolic attractors on surfaces Todd Fisher [email protected] Department

The structure ofhyperbolic attractors on

surfacesTodd Fisher

[email protected]

Department of Mathematics

University of Maryland, College Park

The structure of hyperbolic attractors on surfaces – p. 1/21

Outline of TalkBackground

Examples

Previous Results

Outline of Argument and open questions



Examples

Previous Results




Examples

Previous Results




Examples

Previous Results



StandingAssumptions

M is a compact smooth boundaryless surface.

f ∈ Diff(M)


Transitivity andMixing

Definition: A set X is transitive for a map f if thereis a point x ∈ X with a dense forward orbit

If M is a compact manifold and f adiffeomorphism of M , then equivalently we saythat given two open sets U and V in X ∃ n ∈ N

such that fn(U) ∩ V 6= ∅.

Definition: A set X is mixing (topologically) if forany open sets U and V in X ∃ N ∈ N such that ∀n ≥ N fn(U) ∩ V 6= ∅.














Hyperbolic Attractors

Definition: A compact invariant set Λ forf ∈ Diff(M) is hyperbolic if the tangent spacehas a continuous invariant splittingTΛM = E

s ⊕ Eu where E

s is uniformly contractingand E

u is uniformly expanding.

Definition: A set X is an attractor for a map f if ∃neighborhood U (an attracting set) of X suchthat X =

⋂

n∈Nfn(U) and f(U) ⊂ U .

Definition: A set Λ is a hyperbolic attractor if Λ istransitive and has an attracting set U .




s ⊕ Eu where E




⋂






s ⊕ Eu where E




⋂




Properties ofHyperbolic Attractors

For each x ∈ Λ we know W u(x) ⊂ Λ

If Λ is mixing we know W u(x) = Λ and W s(x) isdense in W s(Λ) ∀ x ∈ Λ.

Λ =⋃k

i=1 Λi where Λi’s are compact disjoint,f(Λi) = Λi+1, and f k(Λi) is a mixing hyperbolicattractor.





Λ =⋃k






Λ =⋃k



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

int(Ri) ∩ int(Rj) = ∅ if i 6= j

for some ε sufficiently small Ri is(W u

ε (x) ∩ Ri) × (W sε (x) ∩ Ri)

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

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


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


ε (x) ∩ Ri) × (W sε (x) ∩ Ri)






ε (x) ∩ Ri) × (W sε (x) ∩ Ri)






ε (x) ∩ Ri) × (W sε (x) ∩ Ri)




General QuestionThe structure of hyperbolic attractors on surfaceshas been studied extensively by Plykin, Bonatti,Williams, Zhirov, Grines, F. and J.Rodriquez-Hertz, and others.

In general these address the question where weassume we have an attractor, then what can besaid about the set and when are two setshomeomorphic.

Question: Suppose we know the topology of Λand is hyperbolic, what can be concluded aboutthe set?










Statement of MainResult

Theorem 1:(F.) If M is a compact surface and Λ isa nontrivial mixing hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then Λ is either anontrivial mixing hyperbolic attractor or anontrivial mixing hyperbolic repeller for g.

A nontrivial attractor means not the orbit of aperiodic sinkCounterexamples in higher dimensions




A nontrivial attractor means not the orbit of aperiodic sink

Counterexamples in higher dimensions




A nontrivial attractor means not the orbit of aperiodic sinkCounterexamples in higher dimensions


Non-mixing Case

Theorem 2:(F.) If M is a compact surface and Λ isa nontrivial hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then there exists ann ∈ N and sets Λ1, ..., ΛN where Λ =

⋃Ni=1 Λi,

Λi ∩ Λj = ∅ if i 6= j, and each Λi is a mixinghyperbolic attractor or repeller for gn.


Commuting Diffeo.r ≥ 2

Theorem 3:(F.) Let M be a compact surface and Λis a nontrivial hyperbolic attractor forf ∈ Diffr(M), r ≥ 2. Then there exists aneighborhood U of f in Diffr(M) and an openand dense set U ′ ⊂ U such that for all f ′ ∈ U ifg ∈ Diff1(M) where f ′g = gf ′ (g in the centralizerof f ), then g|W s(Λ) = (f ′)j for some j ∈ Z.

This is an extension of a result from the work ofPalis and Yoccoz.


DA example

Let f : T2 → T

2 Anosov map from map

A =

[

2 1

1 1

]

, p a hyperbolic fixed point of f .

pV

p

p

1

2

Let Λ =⋂

n∈Nfn(T2 − V ), then Λ is a hyperbolic

attractor.


DA example

Let f : T2 → T


A =

[

2 1

1 1

]


p

pV

p

p

1

2

Let Λ =⋂


attractor.


DA example

Let f : T2 → T


A =

[

2 1

1 1

]


pV

p

p

1

2

Let Λ =⋂


attractor.


DA example

Let f : T2 → T


A =

[

2 1

1 1

]


pV

p

p

1

2

Let Λ =⋂


attractor.


Plykin Attractor

Another example due to Plykin can be built onthe disk, so embedded into any surface. Note:we need three holes in our domain.


Plykin Attractor


V


Plykin Attractor


V

f(V)


Results of Williamsand Plykin

Williams: Shows that if x is in an attractor on asurface, then a neighborhood of x in the attractoris [0, 1] × C for a Cantor set C. Also, usessymbolic dynamics to classify when twodiffeomorphisms restricted to two attractors areconjugate.

Plykin: Shows trapping region is homeomorphicto disk with holes, then must have at least 3holes.


Results of Williamsand Plykin

Williams: Shows that if x is in an attractor on asurface, then a neighborhood of x in the attractoris [0, 1] × C for a Cantor set C. Also, usessymbolic dynamics to classify when twodiffeomorphisms restricted to two attractors areconjugate.

Plykin: Shows trapping region is homeomorphicto disk with holes, then must have at least 3holes.


Zhirov and F. and J.Rodriquez-Hertz

Zhirov: Classifies when 2 diffeomorphisms areconjugate in neighborhood of attractor. Givesalgorithm for the classification.

F. and J. Rodriquez-Hertz: Obtain a local dynamicaland topological picture of expansive attractors onsurfaces.


Zhirov and F. and J.Rodriquez-Hertz

Zhirov: Classifies when 2 diffeomorphisms areconjugate in neighborhood of attractor. Givesalgorithm for the classification.F. and J. Rodriquez-Hertz: Obtain a local dynamicaland topological picture of expansive attractors onsurfaces.


Outline of argument -2 cases

W u(x, g) = W u(x, f) or W s(x, g) = W u(x, f)for all x ∈ Λ

W u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f)for some x ∈ Λ


Outline of argument -2 cases

W u(x, g) = W u(x, f) or W s(x, g) = W u(x, f)for all x ∈ Λ

W u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f)for some x ∈ Λ


Case 1Assume W u(x, f) = W u(x, g) We show Λ isattractor: Take ε > 0 sufficiently small andV =

⋂

x∈Λ W sε (x) is attracting set. i.e. -

Λ =⋂

n≥0 gn(V ).

Since cl(W u(x, f)) = Λ then can show Λ mixinghyperbolic attractor.


Case 1Assume W u(x, f) = W u(x, g) We show Λ isattractor: Take ε > 0 sufficiently small andV =

⋂

x∈Λ W sε (x) is attracting set. i.e. -

Λ =⋂

n≥0 gn(V ).

Since cl(W u(x, f)) = Λ then can show Λ mixinghyperbolic attractor.


Markov PartitionTheorem

Theorem 4:(F.) If Λ is a hyperbolic set and V is aneighborhood of Λ, then there exists a hyperbolicset Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .

For x, y ∈ Λ̃ in same rectangle W sε (x) ∩ W u

ε (y) isone point in Λ̃. So product structure for points insame rectangle.


Markov PartitionTheorem

Theorem 4:(F.) If Λ is a hyperbolic set and V is aneighborhood of Λ, then there exists a hyperbolicset Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .

For x, y ∈ Λ̃ in same rectangle W sε (x) ∩ W u

ε (y) isone point in Λ̃. So product structure for points insame rectangle.


Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).



x

s

u

uW (x,f)

W (x,g)

W (x,g)



y

x

s

u

uW (x,f)

W (x,g)

W (x,g)



x’

y

x

s

u

uW (x,f)

W (x,g)

W (x,g)



x

s

u

uW (x,f)

W (x,g)

W (x,g)



y

x

s

u

uW (x,f)

W (x,g)

W (x,g)



x

s

u

uW (x,f)

W (x,g)

W (x,g)



x

s

u

uW (x,f)

W (x,g)

W (x,g)


ContradictionSo Λ̃ has interior. Then contains hyperbolicattractor and repeller.

Since U arbitrarily small we show this implies Λhas attractor and repeller

Using density of W u(x, f) in Λ we seecontradiction.










Open Questions

Does the main Theorem hold for codimensionone attractors?

If a hyperbolic set is locally maximal (isolated)for f and hyperbolic for g is it necessarilylocally maximal for g?


Open Questions

Does the main Theorem hold for codimensionone attractors?

If a hyperbolic set is locally maximal (isolated)for f and hyperbolic for g is it necessarilylocally maximal for g?


Documents

The structure of hyperbolic attractors on surfacestfisher/documents/presentations/...The structure of hyperbolic attractors on surfaces Todd Fisher [email protected] Department