Ergodic theorems for set maps under weak forms of additivity

TL;DR

This paper develops ergodic theorems for set maps into Banach spaces under relaxed additivity conditions, connecting asymptotic, almost, and weak forms of additivity, and applies these to amenable group representations.

Contribution

It establishes equivalences among different weak additivity notions and derives new ergodic theorems for amenable groups in Banach spaces, simplifying existing proofs.

Findings

Equivalence of asymptotic and almost additivity under certain conditions

New ergodic theorems for non-additive set maps in Banach spaces

Reduction of non-additive to additive settings for proofs

Abstract

We investigate various relaxations of additivity for set maps into Banach spaces in the context of representations of amenable groups. Specifically, we establish conditions under which asymptotically additive and almost additive set maps are equivalent. For Banach lattices, we further show that these notions are related to a third weak form of additivity adapted to the order structure of the space. By utilizing these equivalences and reducing non-additive settings to the additive one by finding suitable additive realizations, we derive new non-additive ergodic theorems for amenable group representations into Banach spaces and streamline proofs of existing results in certain cases.