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

TL;DR

This paper investigates the complex dynamics of self-maps of the unit disk, focusing on the parabolic case, and explores the potential extension of classification and geometric properties to higher dimensions.

Contribution

It analyzes the parabolic case in one dimension and discusses the challenges of extending these concepts to multi-dimensional settings.

Findings

Parabolic maps are classified into zero-step and non-zero-step cases.

Geometric properties of forward iterates can be generalized to higher dimensions.

The extension of the classification to higher-dimensional parabolic maps remains an open question.

Abstract

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.