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 Hénon maps. In particular, we construct the positive and negative dynamical Green's functions $G_{\mathscr{G}}^\pm$ and the corresponding dynamical Green's currents $μ_{\mathscr{G}}^\pm$ 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 $μ_{\mathscr{G}}^+$ 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 Hénon 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 $\mathbb{C}^2.$
Dynamics of semigroups of entire maps of $\mathbb{C}^k$
本文研究了$\mathbb{C}^k,\; k \ge 2$的一类全纯自同态的Fatou集和朱利亚集的一些基本性质.我们特别感兴趣的研究这些集的半群生成的各种类的全纯自同态的$\mathbb{C}^k,\; k \ge 2.$我们证明了，如果朱利亚集的半群$G$生成的最大通秩$k$在$\mathbb{C}^k$的自同态包含一个孤立点，那么$G$必须包含一个元素，共轭的上三角自同构的$\mathbb{C}^k.$这推广了Fornaess-Sibony定理。其次，我们定义了半群的回归域，并在一定条件下给出了这类域的刻画。
Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles
