Dynamical Properties of Some Non-stationary, Non-simple Bratteli-Vershik Systems

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  • March 20, 2019
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  • Bailey, Sarah
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • Bratteli-Vershik systems, also called adics, are dynamical systems defined on the infinite path space of a Bratteli diagram. We introduce a family adics called limited scope adics for which the number of vertices increases by a constant at each level, and a subfamily determined by positive integer polynomials. We show that the dimension groups of the Bratteli diagrams associated to limited scope adics are intrinsically linked to the dynamics, generalizing a result for Cantor minimal systems, and we explicitly compute them for the subfamily of adics determined by a positive integer polynomials. We show that certain limited scope adics are isomorphic to subshifts. For the systems determined by positive integer polynomials we show that the set of fully-supported invariant ergodic probability measures consists of a one-parameter family of Bernoulli measures. We also show that the systems associated to positive integer polynomials are loosely Bernoulli. A particular limited scope adic system is the Euler adic system, for which the number of paths from the root vertex to a vertex (n,k) is the Eulerian number A(n,k). We show that this system has a unique fully-supported invariant ergodic probability measure and that the system is totally ergodic and loosely Bernoulli.
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  • In Copyright
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  • Petersen, Karl Endel
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  • Open access
Date uploaded
  • October 20, 2010

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