Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Abstract

If F is an exact symplectic map on the d-dimensional cylinder T d ×R d , with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow φ * on the space of shift-invariant probability measures on the space of configurations (R d ) Z . Stationary points of φ * are invariant measures of F , and the rotation vector and all spectral invariants are invariants of φ * . Using φ * and the minimisation technique, we construct minimising measures with an arbitrary rotation vector ρ ∈ R d , and with an additional assumption that F is strongly monotone, we show that the support of every minimising measure is a graph of a Lipschitz function. Using φ * and the relaxation technique, assuming a weak condition on φ * (satisfied e.g. in Hedlund's counter-example, and in the anti-integrable limit) we show existence of double-recurrent orbits of F (and F -ergodic measures) with an arbitrary rotation vector ρ ∈ R d , and action arbitrarily close to the minimal action A(ρ).