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Ergodic Variational Methods: New Techniques and New Problems

In: Proceedings of the International Congress of Mathematicians

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  • Ricardo Mañé

    (I.M.P.A.)

Abstract

Our subject will be minimizing measures of Lagrangian dynamical systems. This is a class of invariant probabilities μ, of the flow generated by the Euler-Lagrange equation associated to a periodic Lagrangian on a closed manifold, selected by the property of minimizing the μ-average of the Lagrangian among all the invariant probabilities with a given asymptotic cycle (in the sense of Schwartzmann [S]). This concept was introduced by Mather [Ma 2] in a successful attempt to produce an analog of the Aubry-Mather theory for systems with more than one degree of freedom. His results on minimizing measures, when applied to periodic Lagrangians on the circle, recover the main results of the Aubry-Mather theory of twist maps. The main concepts and results of Mather’s work will be recalled below.

Suggested Citation

  • Ricardo Mañé, 1995. "Ergodic Variational Methods: New Techniques and New Problems," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 1216-1220, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_115
    DOI: 10.1007/978-3-0348-9078-6_115
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