Olena Ostapyuk | Searcharxiv

arXiv2013

On convergence to the Denjoy-Wolff point in the parabolic case

Based on dynamical behavior, all self-maps of the unit disk in the complex plane can be classified as elliptic, hyperbolic or parabolic. The parabolic case is the most complicated one and branches into two subcases - zero-step and non-zero-step cases. In several dimensions, zero-step and non-zero step cases can be defined for sequences of forward iterates, but it is not known yet if the classification can be extended to parabolic maps of the ball. However, some geometric properties of the forward iterates can be generalized to higher-dimensional case.

math.CV↗

arXiv2010

Backward iteration in the unit ball

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.

math.CV↗