Complex structures of non-autonomous basins

Abstract

We study non-autonomous basins of $F_j$ automorphisms of ${\mathbb{C}}^2$ satisfying $$a_j\Vert p\Vert \le \Vert F_j(p)\Vert \le b_j\Vert p\Vert (\forall p\in \mathbb{B})$$ where $0<a_j<b_j<b<1$. We show that if $b_j^m<a_j$ for some integer $m$ and all $j>0$ then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have ${\prod }_{i=1}^{j-1}{b}_i\le b_j^m$ then the basin is biholomorphic to ${\mathbb{C}}^2$. For the quadratic Hénon family $$F_n(z,w)=(z^2+c^{q_n}w,c^{q_n}z),0<c<1,$$ we study how the asymptotic behavior of the sequence $q_n\to \infty$ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to ${\mathbb{C}}^2$ or is a Short ${\mathbb{C}}^2$, thereby isolating possible sequences for which other complex structures may occur.