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On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by Hénon type maps

Andres Enrique Quintero Santander

TL;DR

The paper studies holomorphic dynamics of a rank-2 free group $G=\langle h_1,h_2\rangle$ generated by Hénon-type maps on ${\mathbb C}^2$, with $h_1(x,y)=(y,P(y)-\delta x)$ and $h_2=R_\theta^{-1}\circ h_1\circ R_\theta$ to induce a nontrivial free-group action. It establishes a non-empty Fatou set for $G$, and develops a random-dynamics/ergodic-theory framework to analyze stationary measures, proving that any $\nu$-stationary measure (with $\nu$ supported on the four generators and their inverses) has compact support; under certain conditions (e.g., disjoint filled Julia sets $K(h_1)$ and $K(h_2)$) stationary measures cannot exist. The results connect topological dynamics with stochastic processes on groups via Kesten’s criterion and the random ergodic theorem, illustrating the rich phenomena that arise from holomorphic group actions on non-compact spaces. The work also outlines extensions to related conjugations and highlights directions for future generalizations.

Abstract

We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon 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.

On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by Hénon type maps

TL;DR

The paper studies holomorphic dynamics of a rank-2 free group generated by Hénon-type maps on , with and to induce a nontrivial free-group action. It establishes a non-empty Fatou set for , and develops a random-dynamics/ergodic-theory framework to analyze stationary measures, proving that any -stationary measure (with supported on the four generators and their inverses) has compact support; under certain conditions (e.g., disjoint filled Julia sets and ) stationary measures cannot exist. The results connect topological dynamics with stochastic processes on groups via Kesten’s criterion and the random ergodic theorem, illustrating the rich phenomena that arise from holomorphic group actions on non-compact spaces. The work also outlines extensions to related conjugations and highlights directions for future generalizations.

Abstract

We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon 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.
Paper Structure (5 sections, 5 theorems, 12 equations)

This paper contains 5 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Hénon Maps, Complex Dynamics and Fatou-Julia Decomposition
  3. Stationary Measures
  4. Results on Stationary measures for $G$
  5. Conclusion

Key Result

lemma 1

The measure $\mu$ is $\nu$-stationary/ergodic if and only if the measure $\nu^{\mathbb N}\times \mu$ on the space $\Gamma^{\mathbb N}\times X$ is $T$-invariant/ergodic where $T:G^{\mathbb N}\times X\to G^{\mathbb N}\times X$ is the map $T\left((\omega_n)_{n=1}^\infty, x\right):=\left(\sigma(\omega_n

Theorems & Definitions (11)

  • definition 1
  • definition 2
  • definition 3
  • remark 1
  • remark 2
  • definition 4
  • lemma 1: furman
  • lemma 2
  • theorem 1: Kesten
  • lemma 3
  • ...and 1 more