  Using probabilistic methods, we prove new rigidity results for groups and
pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate
regularity class ({\em i.e.} between $C^1$ and $C^2$). In particular, we
demonstrate some generalizations of Denjoy's Theorem and the classical Kopell's
Lemma for Abelian groups. After that, these techniques are applied to the study
of codimension 1 foliations. We obtain for instance several generalized
versions of Sacksteder's Theorem in class $C^1$. We conclude with some remarks
about the stationary measure.
