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

TL;DR

This paper develops a novel functional-analytic approach to establish limit theorems and statistical stability for a broad class of piecewise partially hyperbolic maps, including systems with singularities and non-invertibility.

Contribution

It introduces a flexible framework that avoids classical compactness assumptions, enabling explicit convergence rates and applicability to complex dynamical systems.

Findings

Proves exponential decay of correlations.

Establishes Central Limit Theorem for H"older observables.

Provides explicit rates of statistical stability.

Abstract

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are not necessarily continuous nor locally invertible. By employing a flexible functional-analytic framework that bypasses the classical requirement of compact embeddings between Banach spaces, we obtain explicit rates of convergence for the variation of equilibrium states under perturbations. Furthermore, we prove the exponential decay of correlations and the Central Limit Theorem for H\"older observables. A key feature of our approach is its applicability to systems where traditional spectral gap techniques fail due to the presence of singularities and the lack of invertibility. We provide several examples illustrating the scope of our results, including partially hyperbolic attractors over horseshoes, non-invertible dynamics semi-conjugated to Manneville--Pomeau maps, and fat solenoidal attractors.