On the uniform hyperbolicity of some nonuniformly hyperbolic systems

Abstract

Abstract. We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability (probability one with respect to every invariant probability measure) are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

Non-uniform Specification and Large Deviations for Weak Gibbs Measures

“…In fact this is already a consequence from the fact that the Maneville-Pommeau transformation satisfies the (strong) specification property. So, having in mind the results obtained in [AAS03,Cao03] it would be very interesting to answer the following question:…”

Uniform hyperbolicity for random maps with positive Lyapunov exponents

“…We remark that the results are non-trivial even in the special case in which the θ-invariant measure P is a Dirac-δ measure supported on a single fixed point {p}. The setting stated above then reduces to the case in which F : M → M is a standard deterministic dynamical system and an analogous result has been proved in [1,4,5]. The theorem we prove here represents a significant generalization of these results and is obtained by a different argument.…”

Uniform hyperbolicity along periodic orbits

“…There are several plausible ways to answer this question, while some basic ideas go back to pioneering works. Among the examples of the recent works, one can refer to [4,5,10,11,13,18] for the non-existence of zero Lyapunov exponents, [9] for uniform hyperbolicity along periodic orbits with one hyperbolic direction and [14] for uniform hyperbolicity along periodic orbits of a shadowable invariant set.…”