Homoclinic orbits and critical points of barrier functions

Abstract

We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on T n . We also prove a critical point theorem for barrier functions, and the existence of such homoclinic orbits on T 2 as an application.

Generalized characteristics and Lax–Oleinik operators: global theory

“…The study of the local propagation of singularities along generalized characteristics was later refined in [36] and [17]. For weak KAM solutions, local propagation results were obtained in [18] and the Lasry-Lions regularization procedure was applied in [12] to analyze the critical points of Mather's barrier functions. An interesting interpretation of the above singular curves as part of the flow of fluid particles has been recently proposed in [27] (see also [34] for related results).…”

On and beyond propagation of singularities of viscosity solutions

“…As shown in [12], the local propagation of singularities along a Lipschitz curve was studied for viscosity solutions and Mather's barrier functions. In [7], the relations between the critical points of the barrier functions and the homoclinic orbits with respect to Aubry sets was studied. The main methods used in [7] is the combination of Lasry-Lions regularization with standard kernel |x − y| 2 /2t and the critical point theory of mountain pass type.…”

Uniform hyperbolicity of invariant cylinder

“…Although there are infinitely many homoclinic orbits [Z2,CC], it is generic that there is only one minimal homoclinic orbit for each class in g ∈ H 1 (T 2 , Z). As there are countably many homological classes only, the following hypothesis is also generic: (H2): The stable manifold intersects the unstable manifold transversally along each minimal homoclinic orbit.…”