Explicit coupling argument for non-uniformly hyperbolic transformations

Abstract

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map.For nonuniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated to the inducing time.Finally, for nonuniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

Spectral Galerkin methods for transfer operators in uniformly expanding dynamics

“…2 Available theoretical bounds typically scale exponentially with the distortion bound C1 (see [18] and Appendix D). However, at least in the analytic case, the spectrally fast convergence dominates the large theoretical bounds.…”

“…In [18], explicit a priori bounds on decay of correlations were stated in the Lipschitz norm. Specifically, if a map on [0, 1] has expansion coefficient λ and (DD 1 ) distortion constant C 1 , then with V the space of zero-integral functions on [0, 1], the following bound holds:…”

“…# NoteK is row-reduced and so upper-triangular 18 Apply h toû; 19 Right-concatenate h onto H; operator inverse K to solve the linear problem and test for convergence [20,15]. It requires as input only an algorithm to calculate the map f and outputs an estimate for ρ whose error is not rigorously bounded but is of the order of S BV ǫ 1−θ , where ǫ is the floating-point precision and θ is a small number depending on the order of differentiability of f .…”

Rates in Almost Sure Invariance Principle for Dynamical Systems with Some Hyperbolicity

“…Our proof of Lemma 4.3 uses a rather delicate technical adaptation of the argument in [15,Section 4]. It is carried out in Appendix A.…”

Superpolynomial and polynomial mixing for semiflows and flows

“…Various polynomial mixing results in [40,42,61] described in Subsection 1.3 rely on a result of [86] which is formulated only for noninvertible dynamical systems. The extra argument required for passing to invertible systems is due to Gouëzel [54] based on work of [34], and can be found in [65,Appendix B] and [58,Theorem 2.10]. Resolving the corresponding problem for flows turns out to be more subtle (even formulating the correct hypotheses is difficult), and is completed in Bálint et al [14].…”