Aubry set for Asymptotically Sub-Additive Potentials

Given a topological dynamical systems \((X, T)\), consider a sequence of continuous potentials \(F := \{f_n: X → \mathbb{R}\}_{n\geq 1}\) that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set \(M_{\rm max}(F)\) of \(T\)-invariant probabilities that attain the following maximum value \({\rm max} \{\lim_{ n\to\infty} \frac{1}{n} \int f_n d\mu : \mu\ {\rm is}\ T{\rm -invariant\ probability}\}\). For this purpose, we extend the notion of Aubry set, denoted by \(\Omega(F)\). Our main result provide a sufficient condition for the Aubry set to be a maximizing set, i. e., \(\mu\) belongs to \(M_{\rm max}(F)\) if, and only if, its support lies on \(\Omega(F)\). Furthermore, we apply this result to the study of the generalized spectral radius in order to show the existence of periodic matrix configurations approaching this value.