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La décomposition dynamique et la différentiabilité des feuilletages des surfaces. (The dynamic decomposition and the differentiability of foliations of surfaces). (French) Zbl 0596.57019

Let \({\mathcal F}\) be a singular foliation on a compact surface M. In order to analyse the dynamics of \({\mathcal F}\), one can canonically cut up M into subsurfaces bounded by curves transverse to \({\mathcal F}:\) the components of the recurrence of \({\mathcal F}\) (quasiminimal sets) are contained in the ”regions of recurrence” and may be studied separately; on the other hand, the dynamics is trivial in the other regions (”regions of passage”). The paper also offers a definition of a singular foliation of class \(C^ r\) on M, and studies the topological and dynamical features of \(C^ 2\) (or \(C^{\infty})\) foliations.

MSC:

57R30 Foliations in differential topology; geometric theory
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
37C10 Dynamics induced by flows and semiflows
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