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 μ is unique and the support of μ is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus T ^N and v∈R^N, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of and h, the entropy penalized Mather problem min T^N ×R^N L(x,v)dμ(x,v)+S[μ] where the entropy S is given by S[μ] = fT^N×R^Nμ(x, v)lnμ(x, v)/f_RNμ(x, w)dw, and the minimization is performed over the space of probability densities μ(x, v) on ^N×^N that satisfy the discrete holonomy constraint ∫_{N× N} φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μ_{, h} which converges to a Mather measure μ, as , h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim _{, h→0} ln μ_{, h}(A), where A ⊂ ^N×^N. In particular, we prove that the deviation function I can be written as I(x,v)= L(x,v)φ(x)(v)H0, where φ₀ is the unique viscosity solution of the Hamilton Jacobi equation, H(x),x)=H. We also prove a large deviation principle for the limit → 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time AubryMather problem, and present a proof of the existence of a separating subaction.