Complex structures of non-autonomous basins

Abstract

We study non-autonomous basins of $F_j$ automorphisms of $\mathbb C^2$ satisfying $$a_j\|p\|\leq\|F_j(p)\|\leq b_j\|p\|\qquad (\forall p\in \mathbb B)$$ 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 $\prod_{i=1}^{j-1}b_i\leq b_j^m$ then the basin is biholomorphic to $\mathbb C^2$. For the quadratic Hénon family $$F_n(z,w)=\bigl(z^2+c^{q_n}w,c^{q_n}z\bigr), \qquad 0<c<1,$$ 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.