  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 positive half line. 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.
