Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index |
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Authors: | Alberto Abbondandolo Alessio Figalli |
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Institution: | (1) Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 56127 Pisa, Italy;(2) Laboratoire J.-A. Dieudonné, CNRS UMR 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France |
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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, then the asymptotic
Maslov indices of periodic orbits are dense in the half line 0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of
an electromagnetic Lagrangian, then 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.
Dedicated to Vladimir Igorevich Arnold |
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Keywords: | 37J05 37J50 53D12 |
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