# Examples of non-autonomous basins of attraction

## TL;DR

This paper explores examples of non-autonomous basins of attraction in complex spaces, proving biholomorphic equivalences and constructing special types of Short ^k^k^k spaces with various geometric properties, advancing understanding of complex dynamical systems.

## Contribution

It provides new examples of non-autonomous basins of attraction and constructs specific Short ^k^k^k spaces with desired properties, addressing questions related to Bedford's Conjecture.

## Key findings

- Non-autonomous basins from pairs of automorphisms are biholomorphic to ^2.
- Existence of (k-1) disjoint Short ^k^k^k in ^k for k \u2265 3.
- Construction of a Short ^k that contains a Fatou-Bieberbach domain and avoids a codimension 2 algebraic variety.

## 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 $k \geq 3$, 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.