A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere

Marc Bonino¹

¹ Institut Galilée, Département de Mathématiques Université Paris 13 Avenue J.B. Clément 93430 Villetaneuse, France

Fundamenta Mathematicae, Tome 182 (2004) no. 1, pp. 1-40.

Résumé

We provide a topological proof that each orientation reversing homeomorphism of the 2-sphere which has a point of period $k \geq 3$ also has a point of period 2. Moreover if such a $k$-periodic point can be chosen arbitrarily close to an isolated fixed point $o$ then the same is true for the 2-periodic point. We also strengthen this result by proving that if an orientation reversing homeomorphism $h$ of the sphere has no 2-periodic point then the complement of the fixed point set can be covered by invariant open sets where $h$ is conjugate either to the map $(x,y) \mapsto (x+1,-y)$ or to the map $(x,y) \mapsto \frac{1}{2}(x,-y)$.

Détail

DOI : 10.4064/fm182-1-1

Keywords: provide topological proof each orientation reversing homeomorphism sphere which has point period geq has point period moreover k periodic point chosen arbitrarily close isolated fixed point periodic point strengthen result proving orientation reversing homeomorphism sphere has periodic point complement fixed point set covered invariant sets where conjugate either map mapsto y map mapsto frac y