Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions

Asaoka, Masayuki

doi:10.1090/s0002-9939-07-09115-0

Cited by 45 publications

(77 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then the set Λ can play the role of thick horseshoes in Newhouse construction. Geometrical constructions using these kind of "thick" hyperbolic sets provide examples of systems with C 1 -robust heterodimensional cycles 2 (see [3]) or C 1 -robust tangencies (see [28,4]). But these constructions involve quite specific global dynamical configurations, thus they cannot translate to a general setting.…”

mentioning

confidence: 99%

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

Bonatti

Díaz²

2012

Trans. Amer. Math. Soc.

108

View full text Add to dashboard Cite

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...

show abstract

mentioning

confidence: 99%

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

Bonatti

Díaz²

2012

Trans. Amer. Math. Soc.

108

View full text Add to dashboard Cite

show abstract

“…Using this connection, some progress has been made in understanding the symbolic extensions of certain classes of dynamical systems, with particular interest in smooth dynamical systems. For results of this type, see [1,3,4,5,7,10,11,12]. Note that the functions u α are, in general, not harmonic, which stands in stark contrast to most objects of study in ergodic theory (in particular, the entropy function h is harmonic [15,20]).…”

Section: Theorem A4 ([3]mentioning

confidence: 99%

“…Furthermore, the theory of entropy structures and symbolic extensions provides a rigorous description of how entropy emerges on refining scales. Entropy structures and the closely related theory of symbolic extensions [3] have attracted interest in the dynamical systems literature [1,4,5,7,10,11,12], especially with the intention of using entropy structure to obtain information about symbolic extensions for various classes of smooth systems. The purpose of the current work is to investigate a new entropy invariant arising from the theory of entropy structures: the order of accumulation of entropy, which is denoted α 0 (X, T ).…”

mentioning

confidence: 99%

Orders of accumulation of entropy

Burguet

McGoff

2012

Fund. Math.

View full text Add to dashboard Cite

Abstract. For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to M . These bounds are given in terms of the Cantor-Bendixson rank of ex(M ), the closure of the extreme points of M , and the relative Cantor-Bendixson rank of ex(M ) with respect to ex(M ). We also address the optimality of these bounds.

show abstract

“…Therefore all these dynamical systems are asymptotically h-expansive and then admit principal symbolic extensions. On the other hand C 1 maps without symbolic extensions have been built in several works by using generic arguments [17] [1] or with an explicit construction [8].…”

Section: Introductionmentioning

confidence: 99%

“…We say that a map T : M → M defined on a compact manifold is C r with r > 1 when T is [r] times differentiable 1 . The author of this paper built explicit examples [8] proving this upper bound to be sharp.…”

Section: Introductionmentioning

confidence: 99%

Symbolic extensions for nonuniformly entropy expanding maps

Burguet

2010

Colloq. Math.

View full text Add to dashboard Cite

Abstract. We call nonuniformly entropy expanding map any C 1 map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a C r nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A.Maass[14].

show abstract

Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions

Cited by 45 publications

References 10 publications

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

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

Orders of accumulation of entropy

Symbolic extensions for nonuniformly entropy expanding maps

Contact Info

Product

Resources

About