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Sur l’unique ergodicité des 1-formes fermées singulières. (French) Zbl 0561.58024

Invent. Math. (to appear).
We study the relations between the de Rham cohomology class of a closed differential 1-form \(\omega\) with Morse singularities on a manifold M of dimension \(n\geq 3\), and the ergodic properties of the foliation \(F_{\omega}\) it defines. We show by examples that, if the fundamental group is ”large” enough, very different behaviours can occur in the same class. In contrast if the foundamental group admits no surjective homomorphism onto the free group on 3 generators, then \(F_{\omega}\) is always uniquely ergodic provided it has no compact leaf; if the natural homomorphism \(\pi_ 1(M)\to H_ 1(M,Z)/torsion\) does not factor through a free group, then the same result is true in almost every cohomology class. We also give results about the existence of noncompact leaves and the number of ergodic measures.

MSC:

37A99 Ergodic theory
58A12 de Rham theory in global analysis