Random dynamical systems of polynomial automorphisms on $\Bbb{C}^{2}$
Hiroki Sumi
TL;DR
This work studies random dynamical systems generated by polynomial automorphisms on $\mathbb{C}^{2}$, focusing on generalized Hénon maps and their conjugates. By introducing mean stability on $\mathbb{P}^{2}$ minus the line at infinity and employing i.i.d. random iterations with law $\tau$, it proves that generically $\tau$ is mean stable, yielding negative maximal Lyapunov exponents for almost every sequence and initial point, finitely many attracting minimal sets, convergence of orbits to a minimal set, and a spectral gap for the transition operator on Hölder spaces. It also establishes the continuity properties of nonautonomous Julia sets $J_{\gamma}^{\pm}$ and constructs nonautonomous Green functions, leading to zero Lebesgue measure for these Julia sets in typical random regimes. The paper develops new methods for higher dimensional random holomorphic dynamics and demonstrates randomness induced order, density of mean-stable laws, and structural results for minimal sets, with implications for stability under random perturbations and the analysis of nonautonomous Julia sets. Overall, the results provide a comprehensive framework for higher dimensional random holomorphic dynamics and reveal robust mechanisms by which randomness can suppress chaotic behavior.
Abstract
This paper deals with random dynamical systems of polynomial automorphisms (complex generalized Hénon maps and their conjugate maps) of $\Bbb{C}^{2}.$ We show that a generic random dynamical system of polynomial automorphisms has ``mean stablity'' on $\Bbb{C}^{2}$. Further, we show that if a system has mean stability, then (1) for each $z\in \Bbb{C}^{2}$ and for almost every sequence $γ=(γ_{n})_{n=1}^{\infty }$ of maps, the maximal Lyapunov exponents of $γ$ at $z$ is negative, (2) there are only finitely many minimal sets of the system, (3) each minimal set is attracting, (4) for each $z\in \Bbb{C}^{2}$ and for almost every sequence $γ$ of maps, the orbit $\{ γ_{n}\cdots γ_{1}(z) \} _{n=1}^{\infty }$ tends to one of the minimal sets of the system, and (5) the transition operator of the system has the spectrum gap property on the space of Hoelder continuous functions with some exponent. Note that none of (1)--(5) can hold for any deterministic iteration dynamical system of a single complex generalized Hénon map. We observe many new phenomena in random dynamical systems of polynomial automorphisms of $\Bbb{C}^{2}$ and observe the mechanisms. We provide new strategies and methods to study higher-dimensional random holomorphic dynamical systems.