Dynamics of semigroups of H\'{e}non maps

TL;DR

This paper investigates the complex dynamics of semigroups generated by Hénon maps, constructing Green's functions and currents, and analyzing Julia sets and basins of attraction in both autonomous and non-autonomous settings.

Contribution

It introduces a comprehensive framework for understanding the dynamics of semigroups of Hénon maps, including Green's functions, Julia sets, and basins, extending classical results to semigroup and non-autonomous cases.

Findings

Julia sets are unions of individual Julia sets of generators.

Green's currents are supported on the entire Julia set.

Non-autonomous basins are biholomorphic to C2.

Abstract

The goal of this article is two fold. Firstly, we explore the dynamics of a semigroup of polynomial automorphisms of $\mathbb{C}^2$, generated by a finite collection of H\'enon maps. In particular, we construct the positive and negative dynamical Green's functions $G_{\mathscr{G}}^\pm$ and the corresponding dynamical Green's currents $\mu_{\mathscr{G}}^\pm$ for a semigroup $\mathcal{S}$, generated by a collection ${\mathscr{G}}.$ Using them, we show that the positive (or negative) Julia set of the semigroup $\mathcal{S}$, i.e., $\mathcal{J}_{\mathcal{S}}^+$ (or $\mathcal{J}_{\mathcal{S}}^-$) is equal to the closure of the union of individual positive (or negative) Julia sets of the maps, in the semigroup $\mathcal{S}$. Furthermore, we prove that $\mu_{\mathscr{G}}^+$ is supported on the whole of $\mathcal{J}_{\mathcal{S}}^+$ and is also the unique positive closed $(1,1)$-current supported on $\mathcal{J}_{\mathcal{S}}^+$, satisfying a semi-invariance relation that depends on the generating set ${\mathscr{G}}$. Secondly, we study the dynamics of a non-autonomous sequence of H\'{e}non maps, say $\{h_k\}$, contained in the semigroup $\mathcal{S}$. Similarly, as above, here too, we construct the non-autonomous dynamical positive and negative Green's function and the corresponding dynamical Green's currents. Further, we use the properties of Green's function to conclude that the non-autonomous attracting basin of any such sequence $\{h_k\}$, sharing a common attracting fixed point, is biholomorphic to $\mathbb{C}^2.$