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

We investigate the statistical stability of a class of dynamical systems semi-conjugate to pre-piecewise $\mathit{\text{convex}}\mathit{\text{or}}\mathit{\text{expanding}}$ 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 Lüroth maps.