On the number of ergodic minimizing measures for Lagrangian flows

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Abstract

We give an example where for an open set of Lagrangians on the n-torus there is at least one cohomology class c with at least n different ergodic c-minimizing measures. One of the problems posed by Ricardo Mañé in his paper 'Generic properties and problems of minimizing measures of Lagrangian systems' (Nonlinearity, 1996) was the following: Is it true that for generic Lagrangians every minimizing measure is uniquely ergodic? A weaker statement is that for generic Lagrangians every cohomology class has exactly one minimizing measure, which of course will be ergodic. Our example shows that this can't be true and as a consequence one can hope to prove at most that for a generic Lagrangian, for every cohomology class there are at most n corresponding ergodic minimizing measures, where n is the dimension of the first cohomology group.

Original languageEnglish
Pages (from-to)501-507
Number of pages7
JournalDiscrete and Continuous Dynamical Systems
Volume17
Issue number3
StatePublished - 1 Mar 2007

Keywords

  • Action-minimizing measures
  • Lagrangian systems

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