On the uniform hyperbolicity of some nonuniformly hyperbolic systems

By José F. Alves, Vitor Araujo, B. Saussol. 2001

57 citations recorded by the source corpus.

We give sufficient conditions for the uniform hyperbolicity o certain nonuni- formly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly ex- panding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5a5f2f01274dcd995625fddd814d6f94610d927a

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Large Deviations for Non-Uniformly Expanding Maps (2006, methodology): “We are able to obtain large deviation rates as in ( 4 ) for non-uniformly expanding local diffeomorphisms and also for endomorphisms and maps with non-flat singularities and critical points under a condition on the rate of approximation of most orbits to the critical/singular set.”
Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies (2003, background): “…there is a uniformly hyperbolic core in the following sense: there exists a region W containing one point of the orbit of tangency, and there exists constants σ > 1 > λ > 0 such that, for each point p ∈ Λ ∩W that returns to W at the k-th iterate, we have (3) ‖DΦk|Esp‖ ≤ λ n and ‖DΦk|Eup ‖ ≥ σ n.”
Equilibrium states for non-uniformly expanding maps (2003, background): “See [ AAS03 ]. In fact, the same conclusion holds, more generally, if all invariant measures have only positive Lyapunov exponents.”