Ergodic Theory and Dynamical Systems

Research Article

Aperiodic substitution systems and their Bratteli diagrams

S. BEZUGLYIa1, J. KWIATKOWSKIa2 and K. MEDYNETSa3

a1 Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: bezuglyi@ilt.kharkov.ua)

a2 Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn 10561, Poland (email: jkwiat@mat.uni.torun.pl)

a3 Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: medynets@ilt.kharkov.ua)

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.

(Received May 28 2007)

(Revised February 19 2008)