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The entropies of renewal systems

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Abstract

Renewal systems are symbolic dynamical systems originally introduced by Adler. IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX W ⊂A Z formed by bi-infinite concatenations of words fromW. Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Jacob Goldberger

  2. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Douglas Lind

  3. Department of Statistics, Tel Aviv University, Tel Aviv, Israel

    Meir Smorodinsky

Authors
  1. Jacob Goldberger
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  2. Douglas Lind
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  3. Meir Smorodinsky
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Additional information

The authors were supported in part by NFS grants DMS-8706284, DMS-8814159 and DMS-8820716.

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Goldberger, J., Lind, D. & Smorodinsky, M. The entropies of renewal systems. Israel J. Math. 75, 49–64 (1991). https://doi.org/10.1007/BF02787181

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  • DOI: https://doi.org/10.1007/BF02787181

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