Backward iteration in the unit ball

Details Backward iteration in the unit ball Backward iteration in the unit ball

2009-10-28

arxiv.org/abs/0910.5451v4

Mathematics

Complex Variables

32 pages, 3 figures. Changes from the previous version: 1)Theorem 1.8 is now proven in elliptic case. 2)Examples and open ques

Scientific paper

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.

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