Construction of invariant measures of Lagrangian maps: minimisation and relaxation

If F is an exact symplectic map on the{\it d}-dimensional cylinder , with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow on the space of shift-invariant probability measures on the space of configurations . Stationary points of are invariant measures of F, and the rotation vector and all spectral invariants are invariants of . Using and the minimisation technique, we construct minimising measures with an arbitrary rotation vector , 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 and the relaxation technique, assuming a weak condition on (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 , and action arbitrarily close to the minimal action . Received November 4, 1999; in final form July 29, 2000 / Published online April 12, 2001