Fixed points and periodic points of orientation-reversing planar homeomorphisms

Author:
J. P. Boronski

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3717-3722

MSC (2010):
Primary 55M20; Secondary 54F15, 54H25, 58C30

DOI:
https://doi.org/10.1090/S0002-9939-10-10360-8

Published electronically:
April 13, 2010

MathSciNet review:
2661570

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Abstract | References | Similar Articles | Additional Information

Abstract: Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let be an orientation-reversing planar homeomorphism with a continuum invariant (i.e. ). First, suppose there are at least bounded components of that are invariant under . Then there are at least components of the fixed point set of in . This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a -periodic orbit in with . Then there is a 2-periodic orbit in , or there is a 2-periodic component of . The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere . These results generalize to orientation-reversing homeomorphisms of .

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Additional Information

**J. P. Boronski**

Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849

Email:
boronjp@auburn.edu

DOI:
https://doi.org/10.1090/S0002-9939-10-10360-8

Keywords:
Fixed point,
periodic point,
planar homeomorphism,
continuum

Received by editor(s):
August 8, 2009

Received by editor(s) in revised form:
December 31, 2009

Published electronically:
April 13, 2010

Additional Notes:
The author was supported in part by NSF Grant #DMS0634724

Dedicated:
Dedicated to the memory of Professor Andrzej Lasota (1932–2006)

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2010
American Mathematical Society