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.
Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if 0 is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then f has a fixed point of positive index. The second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces. In this paper we investigate area preserving diffeomorphisms of compact surfaces. We are concerned with "rotation vectors" defined in terms of homology which can be associated to the points of the surface. There are two main results. The first is that if f is an area preserving diffeomorphism of a compact surface which is homotopic to the identity and 0 is in the interior of the convex hull of the rotation vectors of f then f has a fixed point of positive index. The second result concerns the mean rotation vector of f. It asserts that if this vector vanishes then f has a fixed point of positive index. A special case of this second result when the surface in question is a surface of genus zero was proved in [F5]. In ?6 we give several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.
Isolated Invariant Sets and the Morse Index
On stable properties of the solution set of an ordinary differential equation Elementary properties of flows The Morse index Continuation Bibliography.