Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Abstract.

If F is an exact symplectic map on the{\it d}-dimensional cylinder $\mathbb{T}^d \times \mathbb{R}^d$, with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow $\phi^*$ on the space of shift-invariant probability measures on the space of configurations $({\mathbb{R}}^d)^{\mathbb{Z}}$. Stationary points of $\phi^*$ are invariant measures of F, and the rotation vector and all spectral invariants are invariants of $\phi^*$. Using $\phi^*$ and the minimisation technique, we construct minimising measures with an arbitrary rotation vector $\rho \in \mathbb{R}^d$, 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 $\phi^*$ and the relaxation technique, assuming a weak condition on $\phi^*$ (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 $\rho \in \mathbb{R}^d$, and action arbitrarily close to the minimal action $A(\rho)$.