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Attractors with irrational rotation number

Published online by Cambridge University Press:  12 December 2011

LUIS HERNÁNDEZ-CORBATO ,
RAFAEL ORTEGA  and
FRANCISCO R. RUIZ DEL PORTAL
LUIS HERNÁNDEZ-CORBATO
Affiliation:
Departamento de Geometría y Topología, Facultad de CC. Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: luishcorbato@mat.ucm.es
RAFAEL ORTEGA
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. e-mail: rortega@ugr.es
FRANCISCO R. RUIZ DEL PORTAL
Affiliation:
Departamento de Geometría y Topología, Facultad de CC. Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: r_Portal@mat.ucm.es
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Abstract

Let h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 59 - 77
Copyright
Copyright © Cambridge Philosophical Society 2011

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