We provide a topological proof that each orientation reversing homeomorphism of the 2-sphere which has a point of period k ≥ 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) ↦ (x+1,-y) or to the map (x,y) ↦ 1/2(x,-y).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-1-1,
author = {Marc Bonino},
title = {A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {1-40},
zbl = {1099.37030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-1-1}
}
Marc Bonino. A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere. Fundamenta Mathematicae, Tome 184 (2004) pp. 1-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-1-1/