Ricardo Bioni | Searcharxiv

arXiv2025

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

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.

math.DS↗

arXiv2025

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

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 $ζ$-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 $δ$, 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 $O(δ^ζ\log δ)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.

math.DS↗