On calibrated and separating sub-actions Eduardo Garibaldi

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 A ̄ 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 ∈ 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(xi) + hA(xi, x) for some xi ∈ Ω(A), where hA(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 (Σ̂, σ̂). ∗Partially supported by CNPq, PRONEX – Sistemas Dinâmicos, Instituto do Milênio, and beneficiary of CAPES financial support. †Partially supported by ANR BLANC07-3 187245, Hamilton-Jacobi and Weak KA