Classifying Hyperbolic Ergodic Stationary Measures on Compact Complex Surfaces with Large Automorphism Groups

Let $X$ be a compact complex surface. Consider a finitely supported probability measure $\mu$ on $\text{Aut}(X)$ such that $\Gamma_{\mu} = \langle \text{Supp}(\mu)\rangle<\text{Aut}(X)$ is non-elementary. We do not assume that $\Gamma_{\mu}$ contains any parabolic elements. In this paper, we study and classify hyperbolic, ergodic $\mu$-stationary probability measures.