 00451 Sinisa Slijepcevic
 Construction of invariant measures of Lagrangian maps: minimisation and relaxation
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Nov 12, 00

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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 shiftinvariant 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 counterexample, and in
the antiintegrable limit)
we show existence of doublerecurrent 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
$A(\rho)$.
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