Application of Infinite Labeled Graphs to Symbolic Dynamical Systems

Matsumoto, Kengo

doi:10.1007/978-0-8176-4789-6_6

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Symbolic Dynamics and Monic Representations1

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2014

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“…The latter, in turn, is a special graph, falling within the graphs called Bratteli diagrams. The Bratteli diagrams in turn serve as useful models, and have a host of applications in symbolic dynamics; see e.g., [Mat11,Hos00]. In fact there is a substantial literature on dynamics in Bratteli diagrams; see e.g., [FO13,Kar12,BK11,HY11].…”

Section: Symbolic Dynamics and Monic Representationsmentioning

confidence: 99%

Monic representations of the Cuntz algebra and Markov measures

Dutkay

Jørgensen

2014

Journal of Functional Analysis

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Abstract. We study representations of the Cuntz algebras ON . While, for fixed N , the set of equivalence classes of representations of ON is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of ON , that have a cyclic vector for the canonical abelian subalgebra. We show that ON has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic S * i -invariant space.

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