Abstract: 

We consider random dynamical systems of polynomial automorphisms (complex generalized H\’{e}non maps and their conjugate maps) of C2. We show that a generic random dynamical system of polynomial automorphisms has “mean stablity” on C2. Further, we show that if a system is mean stable, then (1) for each z∈C2 and for almost every sequence γ=(γn)∞n=1 of maps, the maximal Lyapunov exponent of γ at z is negative, (2) there are only finitely many minimal sets of the system, (3) each minimal set is attracting, and (4) for each z∈C2 and for almost every sequence γ=(γn)∞n=1 of maps, the orbit {γn⋯γ1(z)}∞n=1 tends to one of the minimal sets of the system. Note that none of (1)–(4) can hold for any deterministic iteration dynamical system of a single complex generalized H’{e}non map. Also, we define “nicely absorbing systems”. We show that if a system has mean stability or the support of the measure of the noise has a non-empty interior, then the system is nicely absorbing. We show that if a system is nicely absorbing, then the system has some nice properties which are similar to those of mean stable systems. We observe many new phenomena in random dynamical systems of polynomial automorphisms of C2 and observe the mechanisms. We provide new strategies and methods to study higher-dimensional random holomorphic dynamical systems. For the preprint, see the following. H. Sumi, Random dynamical systems of polynomial automorphisms on C2, https://arxiv.org/abs/2408.03577

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