Existence d'un feuilletage positivement transverse \`a un hom\'eomorphisme de surface

TL;DR

This paper extends Le Calvez's result by demonstrating the existence of a singular topological foliation transverse to a surface homeomorphism's dynamics, even when the lift has fixed points, with singularities at fixed points.

Contribution

It generalizes previous work by constructing a singular foliation for homeomorphisms with fixed points in their lift, broadening the applicability of transverse foliations in surface dynamics.

Findings

Existence of a singular foliation with fixed points as singularities.

Generalization of Le Calvez's theorem to fixed point cases.

Construction of a transverse foliation in more general settings.

Abstract

Let F be a homeomorphism of an oriented surface M that is isotopic to the identity. Le Calvez proved that if F admits a lift without fixed points to the universal covering of M, then there exists a topological foliation of M transverse to the dynamics. We generalize this result to the case where the lift of F has fixed points. We obtain a singular topological foliation whose singularities are fixed points of F. Le Calvez a montr\'e que si F est un hom\'eomorphisme isotope \`a l'identit\'e d'une surface M admettant un rel\`evement au rev\^etement universel n'ayant pas de points fixes, alors il existe un feuilletage topologique de M transverse \`a la dynamique. Nous montrons que ce r\'esultat se g\'en\'eralise au cas o\`u le rel\`evement de F admet des points fixes. Nous obtenons alors un feuilletage topologique singulier transverse \`a la dynamique dont les singularit\'es sont un ensemble ferm\'e de points fixes de F.