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

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.