Open Access
Fall and Winter 2017 Examples of non-autonomous basins of attraction
Sayani Bera, Ratna Pal, Kaushal Verma
Illinois J. Math. 61(3-4): 531-567 (Fall and Winter 2017). DOI: 10.1215/ijm/1534924839


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.


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Sayani Bera. Ratna Pal. Kaushal Verma. "Examples of non-autonomous basins of attraction." Illinois J. Math. 61 (3-4) 531 - 567, Fall and Winter 2017.


Received: 5 December 2017; Revised: 25 May 2018; Published: Fall and Winter 2017
First available in Project Euclid: 22 August 2018

zbMATH: 06932516
MathSciNet: MR3845733
Digital Object Identifier: 10.1215/ijm/1534924839

Primary: 32H02
Secondary: 32H50

Rights: Copyright © 2017 University of Illinois at Urbana-Champaign

Vol.61 • No. 3-4 • Fall and Winter 2017
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