On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by H\'enon type maps

TL;DR

This paper investigates the complex and stochastic dynamics of specific rank 2 free groups generated by Hénon maps, revealing diverse behaviors including the existence and absence of stationary measures, thus illustrating complex phenomena in holomorphic group actions.

Contribution

It provides the first analysis of holomorphic and random dynamics for these free groups, demonstrating the existence of non-empty Fatou sets and constructing examples without stationary measures.

Findings

Fatou set is non-empty for these groups

Stationary measures are supported on a compact set

Examples without stationary measures are constructed

Abstract

We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two H\'enon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a compact set. With some further care this allows us to construct examples having no stationary measures. These examples illustrate the types of phenomena that may arise when studying holomorphic group actions on non-compact manifolds.