A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere
Volume 182 / 2004
Fundamenta Mathematicae 182 (2004), 1-40
MSC: 37E30, 37C25, 37Bxx.
DOI: 10.4064/fm182-1-1
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 $(x,y) \mapsto \frac{1}{2}(x,-y)$.
