Backward iteration in the unit ball

TL;DR

This paper extends the understanding of backward iteration dynamics from one-dimensional unit disk maps to higher-dimensional unit ball maps, identifying boundary repelling fixed points and constructing conjugations to automorphisms.

Contribution

It generalizes key properties of backward iteration sequences and boundary fixed points from the unit disk to the unit ball in higher dimensions, including conjugation constructions.

Findings

Backward iteration sequences with bounded hyperbolic step converge to boundary points.

Boundary repelling fixed points have several notable properties.

Constructed conjugations of maps to automorphisms at isolated boundary points.

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.