Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

TL;DR

This paper investigates the properties of the asymptotic Maslov index in Hamiltonian systems, showing density of indices and existence of measures with prescribed indices under certain conditions.

Contribution

It establishes density results for asymptotic Maslov indices and constructs invariant measures with prescribed indices using a Mather-like approach.

Findings

Asymptotic Maslov indices of periodic orbits are dense in the positive real line.

Every non-negative number can be approximated by indices of periodic orbits converging to an invariant measure.

Existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index.

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 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.