Illinois Journal of Mathematics

Examples of non-autonomous basins of attraction

Sayani Bera, Ratna Pal, and Kaushal Verma

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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