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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
Published electronically: April 13, 2010
MathSciNet review: 2661570
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Abstract: Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let $ h:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be an orientation-reversing planar homeomorphism with a continuum $ X$ invariant (i.e. $ h(X)=X$). First, suppose there are at least $ n$ bounded components of $ \mathbb{R}^2\setminus X$ that are invariant under $ h$. Then there are at least $ n+1$ components of the fixed point set of $ h$ in $ X$. This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a $ k$-periodic orbit in $ X$ with $ k>2$. Then there is a 2-periodic orbit in $ X$, or there is a 2-periodic component of $ \mathbb{R}^2\setminus X$. The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere $ \mathbb{S}^2$. These results generalize to orientation-reversing homeomorphisms of $ \mathbb{S}^2$.

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Additional Information

J. P. Boronski
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849

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

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