newhouse phenomenon, homoclinic tangencies and uniformity of extremal bundles

Newhouse phenomenon, homoclinic tangenciesand uniformity of extremal bundles

Sylvain Crovisier

From Dynamics to ComplexityToronto, 7-11 may 2012

Uniform hyperbolicity

Consider M, compactmanifold, and f ∈ Diff(M).

f is “hyperbolic” iff

– there exist K1, . . . ,Kn,transitive hyperbolic inv.compact sets:

TKi= E s ⊕ Eu,

– any x 6∈⋃

Ki is trapped:there is U open such thatf (U) ⊂ U andx ∈ U \ f (U).

I Non-trivial dynamics decomposes into finitely many pieces.

TKi= E s ⊕ Eu,

– any x 6∈⋃

TKi= E s ⊕ Eu,

– any x 6∈⋃

Non-hyperbolicity (1): heterodimensional dynamics

Abraham-Smale (1970): first robust non-hyperbolic dynamics.

Theorem (Shub, 1971)

There exists an open set U 6= ∅ in Diff(T4) of transitivediffeomorphisms with hyperbolic fixed points of different stabledimension.

I The dynamics still splits into a single piece.

Non-hyperbolicity (1): heterodimensional dynamics

Abraham-Smale (1970): first robust non-hyperbolic dynamics.

Theorem (Shub, 1971)

There exists an open set U 6= ∅ in Diff(T4) of transitivediffeomorphisms with hyperbolic fixed points of different stabledimension.

I The dynamics still splits into a single piece.

Non-hyperbolicity (2): critical dynamics

Theorem (Newhouse, 1970)

There is an open set U 6= ∅ in Diff2(M2) of diffeomorphisms with arobust homoclinic tangency: there is a transitive hyperbolic set Kand x , y ∈ K such that W u(x) and W s(y) have a non-transverseintersection.

Corollary (Newhouse phenomenon, 1974)

Any generic f ∈ U has infinitely many sinks or sources.

I Generalizes in higher dimensions (Palis-Viana), and also indimension d ≥ 3 for the C 1-topology (Bonatti-Dıaz).

I Unknown on surfaces for the C 1-topology.

Classification of differentiable dynamics

Goal. Decompose Diff(M):

I distinguish different persitent global dynamics (phenomena),

I characterize them by local mechanisms.

Conjectures by Smale, Palis, Pujals, Bonatti,...(See Shub’s survey Stability and genericity in 1971.)

Goal. Decompose Diff(M):

I distinguish different persitent global dynamics (phenomena),

I characterize them by local mechanisms.

Conjectures by Smale, Palis, Pujals, Bonatti,...(See Shub’s survey Stability and genericity in 1971.)

# pieces =∞

hyperbolic

heterodimensional and critical

heterodimensional and critical# pieces <∞

# pieces <∞

universal

heterodimensional

Morse - Smale

Birth of sinks through homoclinic tangency

A hyperbolic periodic orbit is sectionaly dissipative if its largestLyapunov exponents λ1, λ2 satisfy

λ1 + λ2 < 0.

Proposition

If a sectionally dissipative periodic orbit has a homoclinic tangency,one can create a sink by a small perturbation.

Corollary

If densely in an open set U ⊂ Diff(M) there exists homoclinictangencies associated to sectionaly dissipative saddles, then theNewhouse phenomenon holds generically in U .

Characterization of the Newhouse phenomenon

Conjecture. Newhouse phenomenon ⇒ homoclinic tangencies.

Theorem (C-Pujals-Sambarino)

For any open set U ⊂ Diff1(M), are equivalent:

– generic diffeomorphisms in U have infinitely many sinks,

– densely in U there exist homoclinic tangencies associated tosectionally dissipative periodic points.

Known results in low dimensions:

dim(M) = 1. The open set H of hyperbolic diffeos is C r -dense.

dim(M) = 2. (Pujals-Sambarino) In T = Diff1(M) \H, homoclinictangencies are dense. (But T is maybe empty.)

Weaker notions of hyperbolicity

Following Hirsch-Pugh-Shub, Brin-Pesin, Mane, Liao,...

1. An invariant set K has a dominated splitting TK M = E ⊕ F ifthere is N ≥ 1 such that for any x ∈ K and u ∈ Ex , v ∈ Fx unitary,

2 ‖Df N .u‖ < ‖Df N .v‖.

2. An invariant bundle F is uniformly expanded if there is N ≥ 1such that for any x ∈ K and v ∈ Fx unitary,

2 < ‖Df N .v‖.

3. An invariant set K is partially hyperbolic if there exists adominated splitting TK M = E ⊕ F and F is uniformly expanded.

I Partial hyperbolicity prevents the existence of sinks; anddomination, the existence of homoclinic tangencies.

2 ‖Df N .u‖ < ‖Df N .v‖.

2 < ‖Df N .v‖.

2 ‖Df N .u‖ < ‖Df N .v‖.

2 < ‖Df N .v‖.

2 ‖Df N .u‖ < ‖Df N .v‖.

2 < ‖Df N .v‖.

2 ‖Df N .u‖ < ‖Df N .v‖.

2 < ‖Df N .v‖.

Dynamics with infinitely many sinks, without domination

Theorem (Pliss, Mane)

If f has infinitely many sinks, (On), one can by C 1-perturbationturn one of them into a sectionally dissipative saddle.

Theorem (P-S, Wen, Gourmelon)

Consider a sequence (On) of hyperbolic periodic orbits with stabledimension 0 < d s < dim(M). Then one of these cases holds:

– after a C 1-perturbation, one On has a homoclinic tangency,

–⋃

On has a dominated splitting E ⊕ F with dim(E ) = d s .

Corollary. If f , C 1-generic, has infinitely many sinks (On), then

– either after C 1-perturbation there exists a sectionallydissipative periodic orbit with a homoclinic tangency,

– or K = lim sup On has a dominated splitting E ⊕ F ,dim(F ) = 1 (and K does not contain any sink).

–⋃

Hyperbolicity of extremal bundles (1): the C 1-generic case

Theorem (C-P-S)

Consider f , C 1 generic, and K, an invariant compact set with adominated splitting TK M = E ⊕ F such that

– dim(F ) = 1,

– K does not contain any sink.

Then, F is uniformly expanded.

I In particular K is not the limit of a sequence of sinks.

Corollary

For any C 1-generic diffeomorphism which exhibits the Newhousephenomenon, a homoclinic tangency associated to a sectionalydissipative orbit can be obtained by C 1-pertrurbation.

Hyperbolicity of extremal bundles (1): the C 1-generic case

Theorem (C-P-S)

Consider f , C 1 generic, and K, an invariant compact set with adominated splitting TK M = E ⊕ F such that

– dim(F ) = 1,

– K does not contain any sink.

Then, F is uniformly expanded.

I In particular K is not the limit of a sequence of sinks.

Corollary

For any C 1-generic diffeomorphism which exhibits the Newhousephenomenon, a homoclinic tangency associated to a sectionalydissipative orbit can be obtained by C 1-pertrurbation.

Hyperbolicity of extremal bundles (2): obstructions

Consider f ∈ Diff(M) and an invariant compact set K with adominated splitting TK M = E ⊕ F .

Question. Which properties prevents F to be unif. expanded?

Known properties.

– K contains a periodic orbit such that F is not unstable,

– K contains a periodic C 1-curve tangent to F ,

– dim(F ) ≥ 2, (for instance F = E c ⊕ Eu).

– f is only C 1. (There are counter-examples conjugated to anAnosov of T2.)

Hyperbolicity of extremal bundles (2): obstructions

Consider f ∈ Diff(M) and an invariant compact set K with adominated splitting TK M = E ⊕ F .

Question. Which properties prevents F to be unif. expanded?

Known properties.

– K contains a periodic orbit such that F is not unstable,

– K contains a periodic C 1-curve tangent to F ,

– dim(F ) ≥ 2, (for instance F = E c ⊕ Eu).

– f is only C 1. (There are counter-examples conjugated to anAnosov of T2.)

Hyperbolicity of extremal bundles (3): after Mane

Theorem (C-P-S)

Consider f ∈ Diff2(M) and an invariant compact set K with adominated splitting TK M = E ⊕ F , dim(F ) = 1, such that

– all periodic orbit in K have an unstable bundle containing F ,

– there is no periodic curve in K tangent to F ,

then F is uniformly expanded.

Strategy.

– Step 1: topological hyperbolicity along F .

– Step 2: existence of a markov box B.

– Step 3: uniform expansion along F .

Theorem (C-P-S)

Previous results:

– Mane (1985): for endomorphisms of the interval,

– Pujals-Sambarino (2000): for surface diffeomorphisms,

– Pujals-Sambarino: when E is uniformly contracted,

– C-Pujals: when E is “thin trapped” and K is totallydisconnected along E .

Theorem (C-P-S)

Strategy.

– Step 1: topological hyperbolicity along F .

– Step 2: existence of a markov box B.

– Step 3: uniform expansion along F .

Topological hyperbolicity along F

Property.Each x ∈ K has a well defined one-dimensional unstable manifoldW u(x) which is a C 1-curve (topologically) contracted by f −1.

It uses:

Theorem (Hirsch-Pugh-Shub)

If K has a dominated splitting E ⊕ F ,there exists a continuous familyof C 1-curves, (γx)x∈K , tangent at Fx ,that is locally invariant.

and a Denjoy-Schwartz argument.

It uses:

Markov boxes: definition

A box B is a union of local unstable manifolds (Jx), x ∈ K thatare bounded by two stable manifolds such that K has non-emptyinterior in K . � It allows to induce on B.

It is Markovian if for each z ∈ B ∩ f −n(B), one has

– f n(Jz) ⊃ Jf n(z). � B sees the expansion along F .

– z is contained in a sub-box B ′ ⊂ B that meets all the curvesJx and f n(B ′) is a union of curves of B.� One can quotient the dynamics along unstable plaques.

� By distortion estimate: if there is a Markov box, F is uniformlyexpanded.

Construction of Markov boxes: the hyperbolic case

In the hyperbolic case:One builds a semi-geometrical Markov partition covering K .

This extends to weaker setting, for instance:F topologically expanded and E is uniformly contracted.

But...if E is not weakly contracted, Markov boxes could not exist!

(For z ∈ B ∩ f −n(B), the dynamics f n could stretch along E .)

Non-uniform Markov boxes (1)

For proving that F is uniformly expanded:

Consider any ergodic measure µ on K and prove that its Lyapunovexponent along F is positive.

I Could we build a Markov partition for µ?

Question. If µ is a hyperbolic measure, does there exist a Markovpartition for the induced dynamics on a Pesin block?

A box B is non-uniformly Markov if the Markov property holds forany return z ∈ B ∩ f −n(B) which is a hyperbolic time:for some uniform C > 0 and λ ∈ (0, 1) we have

‖Df k|E (z)‖ ≤ C .λk , ∀0 ≤ k ≤ n.

I In our setting, we are able to build boxes that are Markov onlyfor most hyperbolic returns.

The reason is that the hyperbolicity constants (C , λ) that areavailable are too weak for contracting the whole B along thebundle E .

A box B is non-uniformly Markov if the Markov property holds forany return z ∈ B ∩ f −n(B) which is a hyperbolic time:for some uniform C > 0 and λ ∈ (0, 1) we have

‖Df k|E (z)‖ ≤ C .λk , ∀0 ≤ k ≤ n.

I In our setting, we are able to build boxes that are Markov onlyfor most hyperbolic returns.

The reason is that the hyperbolicity constants (C , λ) that areavailable are too weak for contracting the whole B along thebundle E .

Criterion for proving the expansion along F .

µ has a uniform positive Lyapunov exponent along E if:

a) there exists a non-uniform Markov box B,

b) between two consecutive returns in B that are hyperbolictimes, F is expanded.

How to get b):

– One can reduce to consider sets K whose proper compactinvariant sets are F -expanding,

– hence, outside B, the bundle F is expanded.

– Between two consecutive hyperbolic times for E , the bundle Fis expanded (by domination E ⊕ F ).

Criterion for proving the expansion along F .

µ has a uniform positive Lyapunov exponent along E if:

a) there exists a non-uniform Markov box B,

b) between two consecutive returns in B that are hyperbolictimes, F is expanded.

How to get b):

– One can reduce to consider sets K whose proper compactinvariant sets are F -expanding,

– hence, outside B, the bundle F is expanded.

– Between two consecutive hyperbolic times for E , the bundle Fis expanded (by domination E ⊕ F ).

newhouse phenomenon, homoclinic tangencies and uniformity of extremal bundles

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