Random dynamical systems of polynomial automorphisms on $\Bbb{C}^{2}$

TL;DR

This paper investigates the behavior of random dynamical systems generated by polynomial automorphisms on ^2, revealing generic stability, properties of minimal sets, and spectral gaps, which differ significantly from deterministic systems.

Contribution

It introduces new results on mean stability and spectral properties of random polynomial automorphisms in ^2, highlighting phenomena absent in deterministic cases.

Findings

Generic random systems exhibit mean stability on ^2.

Almost every orbit tends to a minimal attracting set.

The transition operator has a spectral gap on Hölder spaces.

Abstract

This paper deals with random dynamical systems of polynomial automorphisms (complex generalized H\'{e}non 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 $\gamma =(\gamma _{n})_{n=1}^{\infty }$ of maps, the maximal Lyapunov exponents of $\gamma $ 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 $\gamma $ of maps, the orbit $\{ \gamma _{n}\cdots \gamma _{1}(z) \} _{n=1}^{\infty }$ 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 H\'{e}non 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.