Abstract
We study the relations between the de Rham cohomology class of a closed differential 1-form ω with Morse singularities on a manifoldM of dimensionn≧3, and the ergodic properties of the foliationFω 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 fundamental group admits no surjective homomorphism onto the free group on 3 generators, thenFω is always uniquely ergodic provided it has no compact leaf; if the natural homomorphism π1(M)→H1(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.
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