On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type $ \sigma: \Sigma \to \Sigma $ and a Hölder observable $ A $. In the ergodic optimization model, one is interested in properties of $A$-minimizing probability measures. If $\bar A$ denotes the minimizing ergodic value of $A$, a sub-action $u$ for $A$ is by definition a continuous function such that $A\geq u\circ \sigma-u + \bar A$. We call contact locus of $u$ with respect to $A$ the subset of $\Sigma$ where $A=u\circ\sigma-u + \bar A$. A calibrated sub-action $u$ gives the possibility to construct, for any point $x\in\Sigma$, 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 $\Omega(A)$, the set of non-wandering points with respect to $A$. We prove that, under certain conditions on $\Omega(A)$, any calibrated sub-action is of the form $u(x)=u(x_i)+h_A(x_i,x)$ for some $x_i\in\Omega(A)$, where $h_A(x,y)$ denotes the Peierls barrier of $A$. We also prove that separating sub-actions are generic among Hölder sub-actions. 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 $(\hat \Sigma, \hat \sigma)$.Hamilton-Jacobi et théorie KAM faible : à l'interface des EDP, systèmes dynamiques lagrangiens et symbolique