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Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

Published online by Cambridge University Press:  03 August 2009

JOSÉ ALISTE-PRIETO
JOSÉ ALISTE-PRIETO*
Affiliation:
Departmento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile Laboratorie J.-A. Dieudonné, UMR 6621 CNRS, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email: [email protected])
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Abstract

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincaré’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 565 - 594
Copyright
Copyright © Cambridge University Press 2009

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