Hyperbolicity and exponential convergence of the Lax–Oleinik semigroup

Abstract

For a convex superlinear Lagrangian L : T M → R on a compact manifold M it is known that there is a unique number c such that the Lax–Oleinik semigroup L t + c t : C ( M , R ) → C ( M , R ) has a fixed point. Moreover for any u ∈ C ( M , R ) the uniform limit u ˜ = lim t → ∞ L t u + c t exists. In this paper we assume that the Aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the Aubry set to the exponential rate of convergence of the Lax–Oleinik semigroup.