On calibrated and separating sub-actions

Abstract

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Holder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If Ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ − u + Ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ − u + Ā. A calibrated sub-action u gives the possibility to construct, for any point x e Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.