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 , where Ω is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with , and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc 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 (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.