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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.
We recall the problem posed by Peres and Solomyak in Problems on self-similar and self-affine sets; an update. Progr. Prob.46 (2000), 95–106: can one find examples of self-similar sets with positive Lebesgue measure, but with no interior? The method in Properties of measures supported on fat Sierpinski carpets, this issue, leads to families of examples of such sets.
We consider two dynamical systems associated with a substitution of Pisot type: the usual
-action on a sequence space, and the
-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”) and study the relation between them. In particular, we show that pure discrete spectrum for the
-action implies pure discrete spectrum for the
-action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every
-action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).
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