Ergodic variational methods: New techniques and new problems. (English) Zbl 0848.58035

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. II. Basel: Birkhäuser. 1216-1220 (1995).
In this paper, a minimizing measure is an invariant probability measure for the flow generated by the Euler-Lagrange equation associated to a periodic Lagrangian on a closed manifold, which minimizes the \(\mu\)-average of the Lagrangian among all invariant probabilities within a given asymptotic cycle in the sense of Schwartzmann. This concept had been introduced by J. Mather. The paper concerns mainly minimizing measures for generic Lagrangians. The author states existence, uniqueness and unique ergodicity results, and several open problems. The proofs are only sketched and the author refers to a preprint [‘Generic properties and problems of minimizing measures’, I.M.P.A. (1993)] which will hopefully appear in print.
For the entire collection see [Zbl 0829.00015].


37A99 Ergodic theory
28D05 Measure-preserving transformations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems


Zbl 0799.58030