## Illinois Journal of Mathematics

### 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 (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 $\mathbb{C}^{k}$’s with specified properties. First, we show that for $k\geq3$, 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.

#### Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 531-567.

Dates
Received: 5 December 2017
Revised: 25 May 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924839

Digital Object Identifier
doi:10.1215/ijm/1534924839

Mathematical Reviews number (MathSciNet)
MR3845733

Zentralblatt MATH identifier
06932516

Subjects
Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 32H50: Iteration problems

#### Citation

Bera, Sayani; Pal, Ratna; Verma, Kaushal. Examples of non-autonomous basins of attraction. Illinois J. Math. 61 (2017), no. 3-4, 531--567. doi:10.1215/ijm/1534924839. https://projecteuclid.org/euclid.ijm/1534924839

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