Aubry–Mather theory for homeomorphisms

Abstract In this paper, we develop a variational approach to study the dynamic of a homeomorphism on a compact metric space. In particular, we describe orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semi-distance. We give the link with Auslander’s notion of generalized recurrence, and recover in a different way some parts of a more recent work of Akin and Auslander.