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A hierarchy of topological systems with completely positive entropy

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Abstract

We define a hierarchy of systems with topological completely positive entropy in the context of countable amenable continuous group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.

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Acknowledgments

We thank Nishant Chandgotia for very helpful remarks and prolific discussions concerning this work, and Hanfeng Li for valuable comments on an earlier draft. We are also grateful to anonymous referees for their valuable comments. We also wish to thank PIMS for their support. This research was partially supported by the FONDECYT grant 11200037, ANR projects CoCoGro (ANR-16-CE40-0005) and CODYS (ANR-18-CE40-0007), NSERC (PDF-487919) and CONACyT (287764).

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Authors and Affiliations

  1. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile

    Sebastián Barbieri

  2. CONACyT and Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico

    Felipe García-Ramos

Authors
  1. Sebastián Barbieri
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  2. Felipe García-Ramos
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Correspondence to Sebastián Barbieri.

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Barbieri, S., García-Ramos, F. A hierarchy of topological systems with completely positive entropy. JAMA 143, 639–680 (2021). https://doi.org/10.1007/s11854-021-0167-2

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  • DOI: https://doi.org/10.1007/s11854-021-0167-2

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