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TWO-SIDED SHIFT SPACES OVER INFINITE ALPHABETS

Part of: Topological dynamics Connections with other structures, applications

Published online by Cambridge University Press:  22 March 2017

DANIEL GONÇALVES ,
MARCELO SOBOTTKA  and
CHARLES STARLING
DANIEL GONÇALVES*
Affiliation:
Departamento de Matemática, UFSC, Florianópolis, 88040-900, Brazil email daemig@gmail.com
MARCELO SOBOTTKA
Affiliation:
Departamento de Matemática, UFSC, Florianópolis, 88040-900, Brazil email sobottka@mtm.ufsc.br
CHARLES STARLING
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5, Canada email cstar050@uottawa.ca
*
✉daemig@gmail.com
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Abstract

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Ott, Tomforde and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea, we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets, our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any $M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

Keywords

symbolic dynamicsinfinite alphabet shift spacestwo-sided shift spaces

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 54H20: Topological dynamics 37B15: Cellular automata
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 3 , December 2017 , pp. 357 - 386
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

D. Gonçalves was partially supported by Capes grant PVE085/2012 and CNPq. M. Sobottka was supported by CNPq-Brazil grants 304813/2012-5, 480314/2013-6 and 308575/2015-6; part of this work was carried out while he was a postdoctoral fellow of CAPES-Brazil at the Center for Mathematical Modeling, University of Chile. C. Starling was supported by CNPq; work on this paper occurred while he held a postdoctoral fellowship at UFSC.

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