Elements of a metric spectral theory

TL;DR

This paper introduces a metric-based framework for spectral theory, enabling analysis of non-linear and random systems, with applications to ergodic theorems and surface homeomorphisms.

Contribution

It develops a novel metric spectral theory approach that extends classical linear spectral results to non-linear and stochastic contexts.

Findings

Generalized mean ergodic theorem in metric spaces

Extended Wolff-Denjoy theorem for non-linear maps

Thurston's spectral theorem applied to surface homeomorphisms

Abstract

This paper discusses a general method for spectral type theorems using metric spaces instead of vector spaces. Advantages of this approach are that it applies to genuinely non-linear situations and also to random versions. Metric analogs of operator norm, spectral radius, eigenvalue, linear functional, and weak convergence are suggested. Applications explained include generalizations of the mean ergodic theorem, the Wolff-Denjoy theorem and Thurston's spectral theorem for surface homeomorphisms.