On the number of ergodic minimizing measures for Lagrangian flows

Pages: 501 - 507, Volume 17, Issue 3, March 2007

doi:10.3934/dcds.2007.17.501       Abstract        Full Text (125.7K)       Related Articles

Radu Saghin - Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S2E4, Canada (email)

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.

Keywords:  Lagrangian systems, Action-minimizing measures.
Mathematics Subject Classification:  Primary: 37J50; Secondary: 37C40, 34C40.

Received: September 2005;      Revised: September 2006;      Published: December 2006.