Illinois Journal of Mathematics

Backward iteration in the unit ball

Olena Ostapyuk

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Abstract

We will consider iteration of an analytic self-map $f$ of the unit ball in $\mathbb{C}^{N}$. Many facts were established about such dynamics in the 1-dimensional case (i.e., for self-maps of the unit disk), and we will generalize some of them in higher dimensions. In particular, in the case when $f$ is hyperbolic or elliptic, it will be shown that backward-iteration sequences with bounded hyperbolic step converge to a point on the boundary. These points will be called boundary repelling fixed points and will possess several nice properties. At each isolated boundary repelling fixed point, we will also construct a (semi) conjugation of $f$ to an automorphism via an analytic intertwining map. We will finish with some new examples.

Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1569-1602.

Dates
First available in Project Euclid: 12 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1373636697

Digital Object Identifier
doi:10.1215/ijm/1373636697

Mathematical Reviews number (MathSciNet)
MR2942116

Zentralblatt MATH identifier
1269.30032

Subjects
Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]
Secondary: 32A40: Boundary behavior of holomorphic functions 32H50: Iteration problems

Citation

Ostapyuk, Olena. Backward iteration in the unit ball. Illinois J. Math. 55 (2011), no. 4, 1569--1602. doi:10.1215/ijm/1373636697. https://projecteuclid.org/euclid.ijm/1373636697


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