Une nouvelle preuve du théorème de point fixe de Handel
M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.
Conjugacy invariants for Brouwer mapping classes
We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.
A stable/unstable “manifold” theorem for area preserving homeomorphisms of two manifolds
The stable/unstable manifold theorem for hyperbolic diffeomorphisms has proven to be of extreme importance in differentiable dynamics. We prove a stable/unstable "manifold" theorem for area preserving homeomorphisms of orientable two manifolds having isolated fixed points of index less than 1. The proof relies uponl the concept of ftee mnodification which was first developed by Morton Brown for homeomorphisms of two manifolds and later extended by Pelikan and Slaminka for area preserving homeomorphisms of two manifolds.