Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

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 $(x,y) \mapsto \frac{1}{2}(x,-y)$.