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

Fundamenta Mathematicae (2004)

• Volume: 182, Issue: 1, page 1-40
• ISSN: 0016-2736

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## Abstract

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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).

## How to cite

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Marc Bonino. "A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere." Fundamenta Mathematicae 182.1 (2004): 1-40. <http://eudml.org/doc/283190>.

@article{MarcBonino2004,
abstract = {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).},
author = {Marc Bonino},
journal = {Fundamenta Mathematicae},
keywords = {homeomorphism of the 2-sphere; orientation reversing; periodic point},
language = {eng},
number = {1},
pages = {1-40},
title = {A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere},
url = {http://eudml.org/doc/283190},
volume = {182},
year = {2004},
}

TY - JOUR
AU - Marc Bonino
TI - A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere
JO - Fundamenta Mathematicae
PY - 2004
VL - 182
IS - 1
SP - 1
EP - 40
AB - 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).
LA - eng
KW - homeomorphism of the 2-sphere; orientation reversing; periodic point
UR - http://eudml.org/doc/283190
ER -

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