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

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 as defined in [8]. 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 behavior of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains 12 subset of subset of Cq, where 12 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 D subset of C obtained by Bracci and Poggi-Corradini in [14,27,28]. In particular, with our geometric approach we are able to answer a question, open even for the unit ball Bq subset of Cq (see [5,26]), namely that for holomorphic parabolic self -maps any escaping backward orbit with bounded step always converges to a point in the boundary. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).