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 C k . In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of C 2 of a prescribed form is biholomorphic to C 2 . This, in particular, provides a partial answer to a question raised in [2] in connection with Bedford's Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short C k 's with specified properties. First, we show that for k ≥ 3, there exist (k − 1) mutually disjoint Short C k 's in C k . Second, we construct a Short 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 C k 's with (piece-wise) smooth boundaries.

A transcendental Hénon map with an oscillating wandering Short $\mathbb{C}^2$

“…Instead we will prove that there exist holomorphic automorphisms with wandering Fatou components equivalent to a Short C 2 . It immediately follows that such maps give rise to infinitely many disjoint Short C 2 's, giving an alternative construction to an observation from [BPV17]. Recall that a Fatou component F 0 of a map F is wandering if F k (F 0 ) ∩ F j (F 0 ) = ∅ for all k = j.…”

On the automorphism group of certain Short $\mathbb C^2$'s

“…Much like Fatou-Bieberbach domains, Short C 2 's come in a variety of shapes and sizes, and possess a number of intriguing properties as can be seen from the examples in [1], [5], [6], [7], [8] and [13]. The purpose of this paper is to study the holomorphic automorphism group of Short C 2 's that arise in (ii) above.…”