Periodic points and homoclinic classes

Abdenur, Flavio; Bonatti, Christian; Crovisier, Sylvain; Díaz, Lorenzo J.; Wen, Lan

doi:10.1017/s0143385706000538

Cited by 75 publications

(107 citation statements)

References 25 publications

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“…Then [ABCDW,Lemma 3.4] linearizes the heterodimensional cycle producing an affine heterodimensional cycle. This heterodimensional cycle [ABCDW,Section 3.2] produces, for every large ℓ, m, a periodic point r ℓ,m whose orbit spends exactly ℓ.Π(p) times shadowing the orbit of p and m.Π(q) times shadowing the orbit of q and an bounded time outside a small neighborhood of these two orbits.…”

Section: Periodic Measures In Homoclinic Classes Of C 1 -Generic Diffmentioning

confidence: 99%

Nonuniform hyperbolicity for C 1-generic diffeomorphisms

2011

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We study the ergodic theory of non-conservative C 1 -generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1 -generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1 -generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.In addition, confirming a claim made by R. Mañé in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C 1 .

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Section: Periodic Measures In Homoclinic Classes Of C 1 -Generic Diffmentioning

confidence: 99%

Nonuniform hyperbolicity for C 1-generic diffeomorphisms

2011

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“…In this section, we borrow some arguments and results from [2,9] in order to prove that diffeomorphisms with co-index one heterodimensional cycles yield blender-horseshoes.…”

Section: Co-index One Cycles and Blender Horseshoesmentioning

confidence: 99%

“…Let us observe that, for an open and dense subset of T (M ), a chain recurrence class is either hyperbolic or has index variation, see [2].…”

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confidence: 99%

Abundance of $C^{1}$-robust homoclinic tangencies

Bonatti

Díaz²

2012

Trans. Amer. Math. Soc.

108

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A diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 1 -neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic set Λ g , depending continuously on g, such that the stable and unstable manifolds of Λ g have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with C 1 -robust homoclinic tangencies.Using blender-horseshoes, we prove that homoclinic classes of C 1 -generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display C 1 -robust homoclinic tangencies.keywords: chain recurrence set, dominated splitting, heterodimensional cycle, homoclinic class, homoclinic tangency, hyperbolic set.MSC 2000: 37C05, 37C20, 37C25, 37C29, 37C70.1 Introduction Framework and general settingA homoclinic tangency is a dynamical mechanism which is at the heart of a great variety of non-hyperbolic phenomena: persistent coexistence of infinitely many sinks [22], Hénon-like strange attractors [5,20], super-exponential growth of the number of periodic points [19], and non-existence of symbolic extensions [15], among others. Moreover, homoclinic bifurcations (homoclinic tangencies and heterodimensional cycles) are conjectured to be the main source of non-hyperbolic dynamics (Palis denseness conjecture, see [23]).In this paper, we present a local mechanism generating C 1 -robust homoclinic tangencies. Using this construction, we show that the occurrence of robust tangencies is a quite general phenomenon in the non-hyperbolic setting, specially when the dynamics does not admit a suitable dominated splitting. * This paper was partially supported by CNPq, Faperj, and PRONEX (Brazil) and the Agreement in Mathematics Brazil-France. We acknowledge the warm hospitality of I.M.P.A, Institute de Mathématiques de Bourgogne, and PUC-Rio during the stays while preparing this paper 1 Let us now give some basic definitions (in Section 2, we will state precisely the definitions involved in this paper). A transitive hyperbolic set Λ has a homoclinic tangency if there is a pair of points x, y ∈ Λ such that the stable leaf W s (x) of x and the unstable leaf W u (y) of y have some non-transverse intersection Given a hyperbolic set Λ of a diffeomorphism f , for g close to f , we denote by Λ g the hyperbolic set of g which is the continuation of Λ (i.e., Λ g is close to Λ and the dynamics of f on Λ and g on Λ g are conjugate). Definition 1.1 (Robust cycles).• Robust homoclinic tangencies: A transitive hyperbolic set Λ of a C r -diffeomorphism f has a C r -robust homoclinic tangency if there is a C r -neighborhood N of f such that for every g ∈ N the continuation Λ g of Λ for g has a homoclinic tangency.• Robust heterodimensional cycles: A diffeomorphism f has a C r -robust heterodimensional cycle if there are transitive hyperbolic sets Λ and Σ of f whose stable bundles have different dimensions and a C r -n...

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“…(ii) There is a dominated splitting Proof. This is [1,Corollary 3] taking into account that a result by Gourmelon guarantees that the homoclinic tangency can be associated to a saddle inside the homoclinic class (see [19,Corollary,6.6.2, Theorem 6.6.8]). Remark 3.3.…”

Section: Generic Assumptions (See [1 §21])mentioning

confidence: 99%

“…Remark 3.3. In Theorem 3.2 we cannot assure that E is contracting and G is expanding unless the homoclinic class is isolated (see [1,4]). …”

Section: Generic Assumptions (See [1 §21])mentioning

confidence: 99%