00-451 Sinisa Slijepcevic

Construction of invariant measures of Lagrangian maps: minimisation and relaxation (103K, latex2e) Nov 12, 00

Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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 $A(\rho)$.

Files: 00-451.src( 00-451.keywords , lagmap9.TEX )