Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps

Abstract

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $\hat{I}\P$-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $\hat{I}´$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(\hat{I}{´}^{\hat{I}}\P \log \hat{I}´)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.

Previous version: Many improvements in the text have been implemented. Lemmas and corollaries that help to understand the proofs have been added. The text has been restructured, and an important example has been included, where all the required hypotheses were explicitly verified; Current version: We corrected some citations