Quantitative Statistical Stability for the Equilibrium States of Piecewise Partially Hyperbolic Maps

Rafael Bilbao, Ricardo Bioni, Rafael Lucena

Published 2020-08-13Version 1

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We apply the spectral gap property and the $\zeta$-H\"older regularity of the disintegration of its equilibrium state to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $\delta$, we show that the physical measure varies continuously with respect to a strong $L^\infty$-like norm. Moreover, we prove that its modulus of continuity is $O(\delta^\zeta \log \delta)$.