On calibrated and separating sub-actions

Abstract

Consider a transitive expanding dynamical system $\sigma :\Sigma \to \Sigma$, 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 $\Omega (A)$. A separating sub-action is a sub-action such that the sub-cohomological equation becomes an identity just on $\Omega (A)$. 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: ${\hat{\mu }}_j$, $j\in \{1,2,...,l\}$. 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 $\mathbf{x}\in \Sigma$, where ${\mathbf{x}}^i$ is a special point (in the projection of the support of a certain ${\hat{\mu }}_i$) and $h_A$ is the Peierls barrier associated to $A$.