On calibrated and separating sub-actions

Abstract

Consider a transitive expanding dynamical system $ σ: Σ\to Σ$, and a Hölder potential $ A $. In ergodic optimization, one is interested in properties of $A$-maximizing probabilities. Assuming ergodicity, it is already known that the projection of the support of such probabilities is contained in the set of non-wandering points with respect to $ A $, denoted by $ Ω(A) $. A separating sub-action is a sub-action such that the sub-cohomological equation becomes an identity just on $ Ω(A) $. For a fixed Hölder potential $ A $, we prove not only that there exists Hölder separating sub-actions but in fact that they define a residual subset of the Hölder sub-actions. We use the existence of such separating sub-actions in an application for the case one has more than one maximizing probability. Suppose we have a finite number of distinct $A$-maximizing probabilities with ergodic property: $ \hat μ_j $, $ j \in \{1, 2, ..., l\} $. Considering a calibrated sub-action $ u $, under certain conditions, we will show that it can be written in the form $$ u (\mathbf x)= u (\mathbf x^i) + h_A(\mathbf x^i, \mathbf x), $$ for all $ \mathbf x \in Σ$, where $ \mathbf x^i $ is a special point (in the projection of the support of a certain $ \hat μ_i $) and $ h_A $ is the Peierls barrier associated to $ A $.