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Invariant measures on stationary Bratteli diagrams

Published online by Cambridge University Press:  17 July 2009

S. BEZUGLYI ,
J. KWIATKOWSKI ,
K. MEDYNETS  and
B. SOLOMYAK
S. BEZUGLYI
Affiliation:
Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: bezuglyi@ilt.kharkov.ua, medynets@ilt.kharkov.ua)
J. KWIATKOWSKI
Affiliation:
College of Economics and Computer Sciences, Barczewskiego 11, 10106 Olsztyn, Poland (email: jkwiat@mat.uni.torun.pl)
K. MEDYNETS
Affiliation:
Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: bezuglyi@ilt.kharkov.ua, medynets@ilt.kharkov.ua)
B. SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA (email: solomyak@math.washington.edu)
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Abstract

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 973 - 1007
Copyright
Copyright © Cambridge University Press 2009

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