Dynamics of Hilbert nonexpansive maps

TL;DR

This paper explores the dynamics of Hilbert nonexpansive maps within convex domains, highlighting their mathematical properties and applications in nonlinear Perron-Frobenius theory.

Contribution

It provides new insights into the dynamical behavior of Hilbert nonexpansive maps and their role in nonlinear Perron-Frobenius theory.

Findings

Analysis of the contraction properties of Hilbert nonexpansive maps

Connections established between these maps and nonlinear Perron-Frobenius theory

Potential applications in various mathematical and applied contexts

Abstract

In his work on the foundations of geometry, Hilbert observed that a formula which appeared in works by Beltrami, Cayley, and Klein, gives rise to a complete metric on any bounded convex domain. Some decades later, Garrett Birkhoff and Hans Samelson noted that this metric has interesting applications, when considering certain maps of convex cones that contract the metric. Such situations have since arisen in many contexts, pure and applied, and could be called nonlinear Perron-Frobenius theory. This note centers around one dynamical aspect of this theory.