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Aperiodic substitution systems and their Bratteli diagrams

Published online by Cambridge University Press:  01 February 2009

S. BEZUGLYI ,
J. KWIATKOWSKI  and
K. MEDYNETS
S. BEZUGLYI
Affiliation:
Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: bezuglyi@ilt.kharkov.ua)
J. KWIATKOWSKI
Affiliation:
Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn 10561, Poland (email: jkwiat@mat.uni.torun.pl)
K. MEDYNETS
Affiliation:
Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: medynets@ilt.kharkov.ua)
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Abstract

We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 37 - 72
Copyright
Copyright © 2008 Cambridge University Press

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