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Measured foliations and pseudogroups of circle isometries. (Feuilletages mesurés et pseudogroupes d’isométries de cercle.) (French) Zbl 0965.57027
Some results of three papers by G. Levitt [Invent. Math. 88, 635-667 (1987; Zbl 0594.57014); J. Differ. Geom. 31, No. 3, 711-761 (1990; Zbl 0714.57016); Invent. Math. 113, No. 3, 633-670 (1993; Zbl 0791.58055)] are adapted to include the non-orientable case. In more detail, the author deals with codimension one transversally non-orientable foliations \({\mathcal F}\) on a manifold \(M\) and determines the structure of finite-type subgroups \(H\) of two groups: of the fundamental group \(\pi_1(B\Gamma)= \pi_1 (M)/{\mathcal L}\) of Haefliger’s classifying space (where \({\mathcal L}\) is the normal subgroup of loops in the leaves with trivial holonomy) and of the group \(\pi_1 (M)/{\mathcal L}'\) (where \({\mathcal L}'\) is the normal subgroup of loops in leaves). The results are applied to the groups of \(C^2\)-diffeomorphism acting freely on a simply connected 1-manifold, and to the study of orbits of pseudogroups \(\Gamma\) of isometries of the circle \(\mathbb{R}/\mathbb{Z}\) when the definition domains are varied. Especially the function \(e(\Gamma)\) is investigated, where \(e(\Gamma)= \lim\inf \mu(W)\) over all \(W\) that intersect all orbits of \(\Gamma\) (and \(\mu\) is the Lebesgue measure).

57R30 Foliations in differential topology; geometric theory
37E10 Dynamical systems involving maps of the circle