*Accepted Paper*

**Inserted:** 18 jan 2008

**Journal:** Journal of Fixed Point Theory and Applications

**Year:** 2008

**Abstract:**

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold $M$. We show that if $M$ has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, the asymptotic Maslov indices of periodic orbits are dense in the half line $[0,+\infty)$. Furthermore, if the Hamiltonian is the Fenchel dual of an electro-magnetic Lagrangian, every non-negative number $r$ is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index $r$. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather's theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector.

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