Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces
Bas Lemmens, Brian Lins, Roger Nussbaum, and Marten Wortel
TL;DR
This paper extends Denjoy-Wolff theorems to fixed point free, nonexpansive mappings in Hilbert's and Thompson's metric spaces within Banach cones, confirming conjectures and broadening understanding of their dynamics.
Contribution
It establishes new Denjoy-Wolff type theorems for nonexpansive mappings in Hilbert's and Thompson's metrics, confirming key conjectures and extending previous results.
Findings
Proved Denjoy-Wolff theorems for a class of nonexpansive mappings
Confirmed conjectures by Karlsson and Nussbaum
Extended results on linear escape rates of these mappings
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.
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