SELECTION OF CALIBRATED SUBACTION WHEN TEMPERATURE GOES TO ZERO IN THE DISCOUNTED PROBLEM

摘要

Consider T(x) = dx (mod 1) acting on S-1, a Lipschitz potential A : S-1 -> R, 0 < lambda < 1 and the unique function b(lambda ): S-1 -> R satisfying b(lambda)(x) = max(T(y)=x){lambda b(lambda)(y) + A(y)}.We will show that, when lambda -> 1, the function b(lambda) - m(A)/1-lambda converges uniformly to the calibrated subaction V(x) = max(mu is an element of M) integral S(y,x) d mu(y), where S is the Mane potential, M is the set of invariant probabilities with support on the Aubry set and m(A) = sup(mu is an element of M) integral Ad mu.For beta > 0 and lambda is an element of (0,1), there exists a unique fixed point u(lambda,beta) : S-1 -> R for the equation e(u lambda,beta(x)) = Sigma(T(y)())(=x)e(beta A(y)+lambda u lambda,beta(y)). It is known that as lambda -> 1 the family e([u lambda,beta-supu lambda,beta]) converges uniformly to the main eigenfuntion phi(beta) for the Ruelle operator associated to beta A. We consider lambda = lambda(beta), beta(1 - lambda(beta)) -> +infinity and lambda(beta) -> 1, as beta -> infinity. Under these hypothesis we will show that 1/beta (u(lambda,beta) - P(beta A)/1-lambda) converges uniformly to the above V, as beta -> infinity. The parameter beta represents the inverse of temperature in Statistical Mechanics and beta -> infinity means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when beta -> infinity.