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Measure-theoretic complexity of ergodic systems

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Abstract

We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small\(\bar d\)-balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(n k) fork ε ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.

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Authors and Affiliations

  1. Institut de Mathématiques de Luminy, CNRS - UPR 9016 Case 930, 163 avenue de Luminy, F13288, Marseille Cedex 9, France

    Sébastien Ferenczi

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  1. Sébastien Ferenczi
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Correspondence to Sébastien Ferenczi.

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Ferenczi, S. Measure-theoretic complexity of ergodic systems. Isr. J. Math. 100, 189–207 (1997). https://doi.org/10.1007/BF02773640

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  • DOI: https://doi.org/10.1007/BF02773640

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