Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

Abstract

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.

Comments on the cosmic convergence of nonexpansive maps

“…When X is hyperbolic there are many results starting from Wolff-Denjoy showing that cosmic convergence holds, for example [9], see also the discussion in [2]. Let us point out the similar question and conjecture in the case X is a convex set equipped with Hilbert's metric, called the Karlsson-Nussbaum conjecture, see [14] for one of the most recent significant contributions.…”

Comments on the cosmic convergence of nonexpansive maps

“…When X is hyperbolic there are many results starting from Wolff-Denjoy showing that cosmic convergence holds, for example [6], see also the discussion in [1]. Let us point out the similar question and conjecture in the case X is a convex set equipped with Hilbert's metric, called the Karlsson-Nussbaum conjecture, see [11] for one of the most recent significant contributions.…”

Characterizing the metric compactification of $$L_{p}$$ spaces by random measures

“…Several works have confirmed the importance of the metric compactification as a topological and geometric tool for the study of isometry groups [30,32,37,38,39], random walks on hyperbolic groups [8,18], random product of semicontractions [29,28,19,17], Denjoy-Wolff theorems [25,26,9,1,31], Teichmüller spaces 2010 Mathematics Subject Classification. Primary 54D35; Secondary 46E30, 46B20, 60G57.…”