Lyapunov exponents and rates of mixing for one-dimensional maps

Abstract

Abstract. We show that one dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive some power of f is mixing and in particular the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, to the average rate at which typical points start to exhibit exponential growth of the derivative.

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

“…We have stated the two results separately because the local diffeomorphism case is sufficiently interesting on its own and to emphasize the fact that the recurrence condition only applies to the case in which a critical and/or singular set exists. Both theorems extend to arbitrary dimension the results of [4] in which similar results were obtained for one-dimensional maps.…”

Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension

“…There are several possible motivations for the construction of Generalized Markov Partitions; we refer to [3,4] for a detailed discussion and references. We mention here one implication for statistical properties of the maps, which follows from our result and from [7,8].…”

“…We restrict ourselves here to the outline of the main steps of the proof of the theorem; the details will appear in [3] and [4]. We observe first of all that the transitivity assumption implies the existence of a point p with dense preimages, and choose some sufficiently small ball ∆ 0 centred at p. This will be the domain of definition of our induced map.…”

Erratum to: Robust Exponential Decay of Correlations for Singular-Flows

“…Moreover, these points belong to two distinct hyperbolic periodic orbits (of saddle type) of the geometric Lorenz attractor and are away from a neighborhood of the singularity at the origin. Hence, these orbits admit a smooth continuation to all C 1+ε nearby vector fields Y that admit a similar construction of smooth cross-section S Y and induced transformation F Y , following the inductive procedure detailed in [1].…”

“…In this argument it is crucial that all the ingredients in the inductive construction be preserved for Y . In particular, the size of hyperbolic balls obtained from hyperbolic times of the one-dimensional Lorenz transformation f Y , which depend on the rates of expansion of f Y but also on the Hölder exponent of D f Y (see [1]). In its turn, the Hölder exponent of D f Y depends on the smoothness of the strong-stable foliation of the Lorenz attractor.…”