On calibrated and separating sub-actions

Abstract

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)We consider a one-sided transitive subshift of finite type sigma : Sigma -> Sigma and a Holder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If (A) over bar denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A >= u o sigma - u + (A) over bar. We call contact locus of u with respect to A the subset of Sigma where A = u o sigma - u + (A) over bar. A calibrated sub-action u gives the possibility to construct, for any point x is an element of 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 separating sub-actions are generic among Holder sub-actions. We also 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) is an element of Omega (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 ((Sigma) over cap, (sigma) over cap).404577602Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)PRONEX - Sistemas DinamicosInstituto do MilenioCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)ANR [BLANC07-3_187245]Hamilton-JacobiWeak KAM TheoryConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)ANR [BLANC07-3_187245