Random dynamics of plane polynomial automorphisms

TL;DR

This paper studies the dynamics of random products of plane polynomial automorphisms, establishing the existence of Green functions and properties of stationary measures under certain conditions.

Contribution

It introduces the existence of dynamical Green functions for non-elementary groups generated by loxodromic automorphisms and analyzes stationary measures in this context.

Findings

Existence of Green functions for certain random automorphism groups

Stationary measures are shown to be compactly supported

Application of Roda's theorem demonstrates stiffness in non-dissipative cases

Abstract

Let $\mu$ be a finitely supported probability measure on the group of automorphisms of $\mathbb{A}^2_\mathbb{C}$. If the group generated by the support of $\mu$ is non-elementary and contains only loxodromic elements, we show the existence of dynamical Green functions associated to random products. We derive consequences for $\mu$-stationary measures: they are compactly supported, and we can apply Roda's theorem to show stiffness when the action is non-dissipative.