A Livšic-type theorem and some regularity properties for nonadditive sequences of potentials

We study some notions of cohomology for asymptotically additive sequences and prove a Livšic-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 how to obtain almost (and asymptotically) additive sequences of Hölder 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 Hölder continuous function. 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.