Ergodic theorems for set maps under weak forms of additivity
Raimundo Briceño, Godofredo Iommi
TL;DR
The paper develops a unified framework for non-additive ergodic theory by introducing and relating three weak additivity notions—asymptotically additive, almost additive, and Riesz-almost additive—for set maps valued in Banach spaces under amenable-group representations. It proves that asymptotic additivity yields additive realizations and, in residually finite groups, implies almost additivity, enabling reduction to additive ergodic theorems. The authors derive mean and pointwise ergodic theorems for asymptotically additive set maps (and almost additive ones in the residually finite case), and extend these results to sequential settings with constant and non-constant error bounds. The work clarifies the hierarchical structure of weak additivity notions, provides additive realizations to non-additive problems, and streamlines proofs of existing results in broader contexts, including Bochner and Banach-lattice frameworks. Key tools include Følner-convergence, coboundary analysis, and additive realizations $A_F v$ with $A_F v=(1/|F|)\sum_{g\in F}\pi(g^{-1})v$.
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.