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 Ck. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of C2 of a prescribed form is biholomorphic to C2. This, in particular, provides a partial answer to a question raised in (A survey on non-autonomous basins in several complex variables (2013) Preprint) in connection with Bedford’s Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short Ck’s with specified properties. First, we show that for k≥3, there exist (k−1) mutually disjoint Short Ck’s in Ck. Second, we construct a Short Ck, large enough to accommodate a Fatou–Bieberbach domain, that avoids a given algebraic variety of codimension 2. Lastly, we discuss examples of Short Ck’s with (piece-wise) smooth boundaries.