Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} 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 $\{F_δ\}_{δ\in [0,1)}$ and their corresponding invariant measures $\{μ_δ\}$. We provide general conditions ensuring that the unperturbed measure $μ_0$ is statistically stable, meaning the map $δ\mapsto μ_δ$ is continuous at $δ= 0$ in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of $μ_δ$ in terms of the perturbation parameter $δ$. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and Lüroth maps.