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

Levitt, Gilbert

Ann. Inst. Fourier 37, No. 3, 85-116 (1987).

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

Keywords:

differentiability; singular foliation on a compact surface; recurrence; quasiminimal sets

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