Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches

TL;DR

This paper studies the statistical stability of dynamical systems semi-conjugate to convex or expanding maps with countably many branches, providing conditions and estimates for the continuity of invariant measures under perturbations.

Contribution

It offers general criteria and explicit bounds for the statistical stability of a broad class of complex dynamical systems with infinite partitions.

Findings

Invariant measures depend continuously on perturbations under certain conditions.

Explicit estimates for the modulus of continuity of invariant measures are derived.

Results apply to systems like Gauss and L"uroth maps with countably many branches.

Abstract

We investigate the statistical stability of a class of dynamical systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches. These systems naturally arise in the study of transformations with unbounded derivatives, discontinuities, or infinite Markov partitions; features that pose significant challenges for stability analysis. Specifically, we consider one-parameter families of transformations $\{F_\delta\}_{\delta \in [0,1)}$ and their corresponding invariant measures $\{\mu_\delta\}$. We provide general conditions ensuring that the unperturbed measure $\mu_0$ is statistically stable, meaning the map $\delta \mapsto \mu_\delta$ is continuous at $\delta = 0$ in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of $\mu_\delta$ in terms of the perturbation parameter $\delta$. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and L\"uroth maps.