THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure mu is unique and the support of mu is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus T-N and v is an element of R-N, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of is an element of and h, the entropy penalized Mather problem min {integral(N)(T) x (N)(R) L(x, v)d mu(x, v) + epsilon S[mu]}, where the entropy S is given by S[mu] = integral T-N x R-N mu(x, v) In mu(x,v)/integral R-N mu(x, w)dw dxdv, and the minimization is performed over the space of probability densities mu(x, v) on TN x RN that satisfy the discrete holonomy constraint integral T-N x R-N phi(x)d mu = 0. It is known [17] that there exists a unique minimizing measure mu(epsilon), h which converges to a Mather measure mu, as epsilon, h -> 0. In the case in which the Mather measure mu is unique we prove a Large Deviation Principle for the limit lim(epsilon,) h -> 0 epsilon In mu(epsilon), h(A), where A subset of T-N x R-N. In particular, we prove that the deviation function I can be written as I(x, v) = L(x, v) + del phi(0)(x) (v) - (H) over bar, where phi(0) is the unique viscosity solution of the Hamilton-Jacobi equation, H (del phi(0)(x) (v) - (H) over bar (0). We also prove a large deviation principle for the limit epsilon -> 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time Aubry-Mather problem, and present a proof of the existence of a separating subaction.