On the hyperbolicity of homoclinic classes (original) (raw)

New Criteria of Generic Hyperbolicity based on Periodic Points

2009

We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a C1C^1C1-open set SU\SUSU then there exists an open and dense subset SAsubsetSU\SA\subset \SUSAsubsetSU of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms. As a simple corollary of our techniques, we have that an arbitrary mathrmC1\mathrm{C}^1mathrmC1-class local diffeomorphism fff of a closed manifold MnM^nMn is uniformly expanding on the closure mathrmClMn(mathrmPer(f))\mathrm{Cl}_{M^n}(\mathrm{Per}(f))mathrmClMn(mathrmPer(f)) of its periodic point set mathrmPer(f)\mathrm{Per}(f)mathrmPer(f), if it is nonuniformly expanding on mathrmPer(f)\mathrm{Per}(f)mathrmPer(f).

On hyperbolic measures and periodic orbits

Discrete and Continuous Dynamical Systems, 2006

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.

Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity

Nonlinearity, 2007

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.

Regularity of Invariant Manifolds for Nonuniformly Hyperbolic Dynamics

Journal of Dynamics and Differential Equations, 2008

For any sufficiently small perturbation of a nonuniform exponential dichotomy, we show that there exist invariant stable manifolds as regular as the dynamics. We also consider the general case of a nonautonomous dynamics defined by the composition of a sequence of maps. The proof is based on a geometric argument that avoids any lengthy computations involving the higher order derivatives. In addition, we describe how the invariant manifolds vary with the dynamics.

Notes on Ergodicity of Partially Hyperbolic Systems

2008

These lecture notes were written to accompany a minicourse given at the School and Workshop on Dynamical Systems at IMPA (Rio de Janeiro), January 2006. Warning: These notes are unedited lecture notes and do contain errors! Proceed with caution... Let f : M → M be a (pointwise) partially hyperbolic diffeomor-phism. Recall what partially hyperbolic means. At every point in M , there exist tangent vectors that are uniformly contracted by the derivative T f and tangent vectors that are uniformly expanded by T f. The contracted vectors lie in an invariant subbundle E s of T M , called the stable subbundle, and the expanded vectors lie in an invariant subbun-dle E u , called the unstable subbundle. The sum of these subbundles E u ⊕ E s is the hyperbolic part of T M. The rest of T M is also under control: there is a T f-invariant complement E c to E u ⊕ E s in T M , called the center bundle. Tangent vectors in E c may be expanded or contracted under the action of T f , but they are neithe...

The embedding flows of C ∞ hyperbolic diffeomorphisms

Journal of Differential Equations - J DIFFERENTIAL EQUATIONS, 2011

In [Weigu Li, J. Llibre, Xiang Zhang, Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows, Amer. J. Math. 124 (2002) 107–127] we proved that for a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) in (Rn,0), if A has a real logarithm with its eigenvalues weakly nonresonant, then F(x) can be embedded in a C∞ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of F(x) and the extension of the Floquet's theory to nonlinear C∞ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result.Next we shall show that the weakly nonresonant condition in the last result on the real logarithm of A is necessary for some C∞ diffeomorphisms F(x)=Ax+f(x) to have C∞ embedding flows.Finally we shall prove that a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) with f(x)=O(|x|2) in (R2,0) has a C∞ ...

A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms

arXiv preprint arXiv:0710.2353, 2007

In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the C 1 topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.

Dynamics near nonhyperbolic fixed points or nontransverse homoclinic points

Mathematics and Computers in Simulation, 2014

We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of the dynamics of center disks, is introduced. Some results on shadowing near a non-hyperbolic fixed point of a homeomorphism are obtained.