Backward iteration in the unit ball

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.