arXiv subjects

Rafael Bilbao

Publications and source records attributed to Rafael Bilbao.

4 recordsLinked to original sources

arXiv2026

Thermodynamic formalism for discontinuous maps and statistical properties of their equilibrium states

In this article, we develop a functional-analytic framework to establish existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states for a broad class of dynamical systems, potentially discontinuous and not necessarily locally invertible. Our approach is applied to a family of piecewise partially hyperbolic maps and associated classes of potentials. We further prove several statistical limit theorems, including exponential decay of correlations, and propose related questions and conjectures. A collection of examples illustrating the applicability of our results is provided, including partially hyperbolic attractors over horseshoes, discontinuous systems, non-invertible dynamical systems admitting a semi-conjugacy to intermittent maps such as the Manneville--Pomeau map, and fat solenoidal attractors.

math.DS↗

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↗

arXiv2020

Uniqueness and Stability of Equilibrium States for Random Non-uniformly Expanding Maps

We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the Thermodynamical Formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

math.DS↗