Markov structures and decay of correlations for non-uniformly expanding dynamical systems

Abstract

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.

Hyperbolic times: frequency versus integrability

“…The integrability properties of this first hyperbolic time map play an important role in the study of some statistical properties of several classes of dynamical systems, such as stochastic stability and decay of correlations; see [2,4,5,9]. The same conclusion of Theorem A can be obtained under the assumption of integrability with respect to Lebesgue measure of the first hyperbolic time map.…”

“…These have become a very useful ingredient in the study of non-hyperbolic dynamical systems, playing an important role in the proof of several results about the existence of absolutely continuous invariant measures and their statistical properties; see [1,2,3,4,5,6]. Ideas of hyperbolic times were implicitly contained in Pesin's theory and in the work of Pliss and Mañé.…”

Large Deviations for Non-Uniformly Expanding Maps

“…Results in [7,29] show that the tail set decays at least sub-exponentially fast, which is not enough to ensure that Corollaries C and E are true for the maps in U ∪Ũ. It is conjectured that the tail set indeed decays exponentially fast and with a uniform rate for all maps in U ∪Ũ.…”

“…Then for the same kind of attracting sets we obtain an upper bound for the subset corresponding to (7). Theorem D. Let f : M → M be a C 2 diffeomorphism exhibiting a partially hyperbolic non-uniformly expanding attracting set Λ with isolating neighborhood U ⊃ Λ.…”

“…[20,61,16,3,5,7,29,10]. The leitmotif is that the sequence {ϕ • f n } n≥0 should behave like an i.i.d.…”