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

TL;DR

This paper establishes quantitative stability results for equilibrium states of piecewise partially hyperbolic maps, showing how invariant measures vary continuously under perturbations with explicit modulus of continuity.

Contribution

It introduces a novel approach using spectral gap and regularity properties to quantify the stability of equilibrium states in piecewise partially hyperbolic systems.

Findings

Invariant measures depend continuously on system perturbations.

Modulus of continuity for measures is $O(\delta^\zeta \log \delta)$ for certain perturbations.

Spectral gap and regularity techniques are effective in stability analysis.

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 $\zeta$-H\"older 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 $\delta$, 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 $O(\delta^\zeta \log \delta)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.