Ergodic Theory and Dynamical Systems

Research Article

Invariant measures on stationary Bratteli diagrams

S. BEZUGLYIa1, J. KWIATKOWSKIa2, K. MEDYNETSa1 and B. SOLOMYAKa3

a1 Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: bezuglyi@ilt.kharkov.ua, medynets@ilt.kharkov.ua)

a2 College of Economics and Computer Sciences, Barczewskiego 11, 10106 Olsztyn, Poland (email: jkwiat@mat.uni.torun.pl)

a3 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA (email: solomyak@math.washington.edu)

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.

(Received July 04 2008)

(Revised April 02 2009)