On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder 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 ε 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 . We prove that separating sub-actions are generic among Hölder sub-actions. We also prove that, under certain conditions on Ω( A ), any calibrated sub-action is of the form u ( x ) = u ( x i ) + h A ( x i , x ) for some x i ∈ Ω( A ), where h A ( x, y ) denotes the Peierls barrier of A . We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type .