TY - JOUR

T1 - On the number of ergodic minimizing measures for Lagrangian flows

AU - Saghin, Radu

PY - 2007/3

Y1 - 2007/3

N2 - 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.

AB - 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.

KW - Action-minimizing measures

KW - Lagrangian systems

UR - http://www.scopus.com/inward/record.url?scp=34247221039&partnerID=8YFLogxK

U2 - 10.3934/dcds.2007.17.501

DO - 10.3934/dcds.2007.17.501

M3 - Article

AN - SCOPUS:34247221039

VL - 17

SP - 501

EP - 507

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 3

ER -