Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

TL;DR

This paper explores the backward dynamics of non-expanding maps in Gromov hyperbolic spaces, introducing stable dilation at boundary fixed points, and applies the theory to holomorphic maps in complex domains, resolving several open problems.

Contribution

It introduces the notion of stable dilation at boundary fixed points and extends dynamical results to holomorphic self-maps of complex domains, addressing open questions in the field.

Findings

Backward orbits with bounded step converge to boundary points.

Stable dilation relates to the dynamical behavior of boundary fixed points.

Results extend to holomorphic maps in complex domains like strongly pseudoconvex and convex finite type.

Abstract

We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $\Omega\subset\subset \mathbb{C}^q$, where $\Omega$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.