Sinisa Slijepcevic
Construction of invariant measures of Lagrangian maps: minimisation and relaxation
(103K, latex2e)

ABSTRACT.  If $F$ is an exact symplectic map on the $d$-dimensional cylinder $\Bbb{T}^d \times \Bbb{R}^d$, 
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 
$({\Bbb{R}}^d)^{\Bbb{Z}}$. 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 $\rho \in \Bbb{R}^d$, 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 the 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 $\rho \in \Bbb{R}^d$, and the action arbitrarily close to the minimal action 
$A(\rho)$.