Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards

Abstract

Gouëzel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.

Poisson Approximations and Convergence Rates for Hyperbolic Dynamical Systems

“…Under similar conditions to the Definition 2.9, [VZ16] proved that µ U (R > n) characterizes the optimal bound for the decay rates of correlations for sufficiently good observables supported on U (see the Theorem 1.3 in [VZ16]); [BMT18] uses operator renewal theory as a method to prove also sharp results on polynomial decay of correlations (see Theorem 3.1 in [BMT18]). For many purposes the aperiodicity in the Definition 2.5 is irrelevant provided the dynamic f : (M, µ) → (M, µ) is mixing (see the Remark 2.2 in [BMT18]). Indeed all dynamical systems, which we consider in applications (section 6), do have a Markov partition.…”

Mixing rates for symplectic almost Anosov maps

“…To obtain upper and lower bounds on mixing rates we further induce T 1 (the first return map of T to Y ) to a two-sided Young tower. This will allows us to apply [3,Theorem 7.4].…”

“…The article [3] requires one more condition. We need to check that there exist C > 0 and θ ∈ (0, 1) such that for every z, z ∈ Q and n ≥ 1, d(T 2 n z, T 2 n z ) ≤ C(θ n + θ s(z,z )−n ).…”

“…On the contrary the study of non-uniformly hyperbolic maps is still unsatisfactory. In particular, few example have been studied [7,11,2,10] and only recently some general strategies are merging [13,3].…”

“…This is the core of the paper and, beside treating the current example, highlights the ingredients needed to obtain such results in more general models. At last, in section 5 we use the estimates of section 4, several facts from [11] and the general theory put forward in [3] to prove Theorem 1.…”