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

Abstract

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 ϵ and h, the entropy penalized Mather problem [Formula: see text] where the entropy S is given by [Formula: see text], and the minimization is performed over the space of probability densities μ(x, v) on 𝕋 N ×ℝ N that satisfy the discrete holonomy constraint ∫ 𝕋 N ×ℝ N φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μ ϵ, h which converges to a Mather measure μ, as ϵ, h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim ϵ, h→0 ϵ ln μ ϵ, h (A), where A ⊂ 𝕋 N ×ℝ N . In particular, we prove that the deviation function I can be written as [Formula: see text], where ϕ 0 is the unique viscosity solution of the Hamilton – Jacobi equation, [Formula: see text]. We also prove a large deviation principle for the limit ϵ→ 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.