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.
Retracts, fixed point index and differential equations
Many problems in the theory of differential equations were initially treated with analytic techniques and later evolved towards more topological approaches. Perhaps the most paradigmatic case is found in the study of nonlinear boundary value problems. The classical proofs based on successive approximations or in the implicit function theorem were soon replaced by the use of fixed points and degree theory. The modern point of view is already found in the famous paper by Leray and Schauder [15]. The same process has been experienced by other branches of differential equations. The next pages are an attempt to illustrate this evolution in two concrete problems. First we will discuss the existence of asymptotic solutions. These are non-trivial solutions tending to the origin as time increases to infinity and they appear in systems of differential equations having the trivial solution. Asymptotic solutions have been studied since Poincaré’s times. The classical method for proving their existence consists in the reduction of the problem to an integral equation. Once this equation has been found one uses the method of successive approximations or the contraction principle. This analytical method leads to the Principle of Linearization and to the Stable Manifold
Geodesics onS2 and periodic points of annulus homeomorphisms
SummaryWe show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric onS2 with positive Gaussian curvature has infinitely many distinct closed geodesics.In this paper we investigate area preserving homeomorphisms of the annulus and their periodic points. The main result is that an area preserving homeomorphism of the annulus which has at least one periodic point (perhaps on the boundary) must in fact have infinitely many interior periodic points.The motivation and main application of this result is the furthering of a program begun by Birkhoff [B] in his book “Dynamical Systems”. There he shows that for many Riemannian metrics onS2, including those with positive curvature, the problem of finding closed geodesics reduces to finding periodic points of a certain area preserving homeomorphism of the annulus. The annulus map in question can be shown to have a periodic point so our main result above can be applied to show the existence of infinitely many distinct closed geodesics whenever this annulus map exists. This is done in Sect. 4 Other quite different approaches to the problem of finding infinitely many geodesics have been successful in handling the cases which do not reduce to the investigation of an annulus homeomorphism (see [Ba]).