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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

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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.

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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