Appears in collection : Foliations and Diffeomorphism Groups / Feuilletages et Groupes de Difféomorphisme

Any (at most countable) family of pairwise transverse (even singular, with saddle or prong singularities) foliations on the plane R² admits a compactification (as the disc D²) by a circle at infinity so that every ray in a leaf tends to a point on the circle, and this compactification is unique up to two natural requirements. Thus every leaf corresponds to a pair of points on the circle. With Th. Barthelmé and K. Mann, we consider the reverse problem and we give a complete answer to the two following questions:

Q1: (realization) Under which hypotheses two sets L⁺, L⁻ of pairs of points on the circle are precisely the pairs of endpoints of leaves of two transverse foliations (we give the answer for singular, and also for nonsingular foliations, and we prove that the foliations are uniquely determined).

More important is the second question: Q2: (completion) Under which hypotheses two sets l⁺,l⁻ or pair of points on the circle correspond the pairs of endpoints of a dense subset of leaves of two transverse foliations (singular or nonsingular, and uniqueness). The uniqueness implies that any group action on the circle preserving l⁺,l⁻ extends in an action on the disc preserving the corresponding foliations. This allows us to prove that if an action G -> Homeo_+(S¹) of a group on the circle is induced by an Anosov-like action G -> Homeo_+(D²), then this action is unique and completely determined by the action on the circle. With Th. Marty, we consider the case of 1 (singular or not) foliation and we give a complete answer to the following questions

Q3: (realization) Under which hypotheses a set L of pairs of points on the circle is precisely the set of pairs of endpoints of leaves of a foliation,

Q4: (completion) Under which hypotheses a set l of pairs of points on the circle corresponds to the pairs of endpoints of a dense subset of leaves of a foliation and we prove again the uniqueness.