A Liv\v{s}ic-type theorem and some regularity properties for nonadditive sequences of potentials

TL;DR

This paper investigates cohomology notions for asymptotically additive sequences, proves a Livšic-type theorem, and characterizes these sequences via equilibrium measures, revealing new examples with unique properties in dynamical systems.

Contribution

It introduces a Livšic-type theorem for almost additive sequences, characterizes them through equilibrium measures, and constructs novel examples of non-equivalent sequences with unique regularity properties.

Findings

Proved a Livšic-type theorem for almost additive sequences.

Characterized almost additive sequences via their equilibrium measures.

Constructed examples of non-equivalent sequences with Hölder continuity and bounded variation.

Abstract

We study some notions of cohomology for asymptotically additive sequences and prove a Liv\v{s}ic-type result for almost additive sequences of potentials. As a consequence, we are able to characterize almost additive sequences based on their equilibrium measures and also show the existence of almost (and asymptotically) additive sequences of H\"older continuous functions satisfying the bounded variation condition (with a unique equilibrium measure) and which are not physically equivalent to any additive sequence generated by a H\"older continuous function. None of these examples were previously known, even in the case of full shifts of finite type. Moreover, we also use our main result to suggest a classification of almost additive sequences based on physical equivalence relations with respect to the classical additive setup.