Examples of non-autonomous basins of attraction

Abstract

The purpose of this paper is to present several examples of non--autonomous basins of attraction that arise from sequences of automorphisms of ${\mathbb{C}}^k$. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of ${\mathbb{C}}^2$ of a prescribed form is biholomorphic to ${\mathbb{C}}^2$. This, in particular, provides a partial answer to a question raised in connection with Bedford's Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short ${\mathbb{C}}^k$'s with specified properties. First, we show that for $k\ge 3$, there exist $(k-1)$ mutually disjoint Short ${\mathbb{C}}^k$'s in ${\mathbb{C}}^k$. Second, we construct a Short ${\mathbb{C}}^k$, large enough to accommodate a Fatou-Bieberbach domain, that avoids a given algebraic variety of codimension $2$. Lastly, we discuss examples of Short ${\mathbb{C}}^k$'s with (piece-wise) smooth boundaries.

29 pages