Hölder regularity and exponential decay of correlations for a class of piecewise partially hyperbolic maps

Abstract

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is ζ-Hölder. We use this fact to obtain exponential decay of correlations on the set of ζ-Hölder functions.

Lipschitz regularity of the invariant measure and statistical properties for a class of random dynamical systems

“…We remark that, Theorem 7.10 is an estimation of the regularity of the disintegration of µ 0 . Similar results, for other sort of skew products are presented in [19], [7], [20].…”

“…The following theorem is an estimate for the Lipschitz constant (see equation ( 27) in Definition 7.2) of the disintegration of the unique F -invariant measure µ 0 in S ∞ . This kind of result has many applications and similar estimations (for other systems) were given in [19], [7] and [20]. In [19], for instance, they use the regularity of the disintegration to prove stability of the F -invariant measure under a kind of ad-hoc perturbation.…”

“…Then 1 = | gdµ| ≤ ||µ|| W . These two facts proves equation (20). Now we are ready to prove Proposition 5.1.…”

“…The regularity of the disintegration was studied in [19], [20] and also in [23] and [7]. In [19], the authors obtained a Bounded Variation regularity, for the disintegration of the physical measure of Lorenz-like maps, for a certain topology.…”

“…This property was used to prove stability of the physical measure of Lorenz-like maps under deterministic perturbations and other statistical results. In [20] the authors proved Holder regularity for the disintegration of the physical measure, for a class of piecewise (possibly with discontinuities) partially hyperbolic maps, semi-conjugated to a non uniformly expanding map. They used this fact to obtain exponential decay of correlations, for Holder observables and others limit theorems.…”