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Prime end rotation numbers of invariant separating continua of annular homeomorphisms


Author: Shigenori Matsumoto
Journal: Proc. Amer. Math. Soc. 140 (2012), 839-845
MSC (2010): Primary 37E30; Secondary 37E45
DOI: https://doi.org/10.1090/S0002-9939-2011-11435-7
Published electronically: November 2, 2011
MathSciNet review: 2869068
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a homeomorphism of the closed annulus $ A$ isotopic to the identity, and let $ X\subset {\rm Int}A$ be an $ f$-invariant continuum which separates $ A$ into two domains, the upper domain $ U_+$ and the lower domain $ U_-$. Fixing a lift of $ f$ to the universal cover of $ A$, one defines the rotation set $ \tilde \rho (X)$ of $ X$ by means of the invariant probabilities on $ X$, as well as the prime end rotation number $ \check \rho _\pm $ of $ U_\pm $. The purpose of this paper is to show that $ \check \rho _\pm $ belongs to $ \tilde \rho (X)$ for any separating invariant continuum $ X$.


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Additional Information

Shigenori Matsumoto
Affiliation: Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda, Surugadai, Chiyoda-ku, Tokyo, 101-8308 Japan
Email: matsumo@math.cst.nihon-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2011-11435-7
Keywords: Continuum, rotation set, prime end rotation number, Brouwer line, foliations
Received by editor(s): December 5, 2010
Published electronically: November 2, 2011
Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research (C) No. 20540096.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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