Aubry set for asymptotically sub-additive potentials

Abstract

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.