Almost-additive ergodic theorems for amenable groups

TL;DR

This paper establishes a general convergence theorem for almost-additive set functions on amenable groups, extending ergodic theory and providing applications to spectral analysis of random operators.

Contribution

It introduces a unified approach to almost-additive ergodic theorems for amenable groups, including Banach space valued results and spectral distribution applications.

Findings

Proves almost-everywhere convergence of additive processes.

Provides a Banach space approximation for spectral distribution functions.

Extends Lindenstrauss ergodic theorem to Banach space valued functions.

Abstract

In this paper we prove a general convergence theorem for almost-additive set functions on unimodular, amenable groups. These mappings take their values in some Banach space. By extending the theory of epsilon-quasi tiling techniques, we set the ground for far-reaching applications in the theory of group dynamics. In particular, we verify the almost-everywhere convergence of abstract approximable bounded, additive processes, as well as a Banach space approximation result for the spectral distribution function (integrated density of states) for random operators on discrete structures in a metric space. Further, we include a Banach space valued version of the Lindenstrauss ergodic theorem for amenable groups.