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Minimal distance between random orbits

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Abstract

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

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Acknowledgements

JR was partially supported by CNPq, by FCT projects PTDC/MAT-PUR/28177/2017 and PTDC/MAT-PUR/4048/2021, with national funds, and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. The third author acknowledges financial support by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by Conselho Nacional de Desenvolvimento Científico - Brasil (CNPq) - PQ 312632/2018-5. SG acknowledges the support of the Centre Henri Lebesgue ANR-11-LABX- 0020-01.

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Authors and Affiliations

  1. IRMAR, CNRS UMR 6625, Université de Rennes, Rennes, 35042, France

    Sébastien Gouëzel

  2. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros, Salvador, 40170-110, Brazil

    Jérôme Rousseau

  3. CREC, Académie Militaire de St Cyr Coëtquidan, Guer, 56381, France

    Jérôme Rousseau

  4. Departamento de Matemática, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos 149, Rio de Janeiro, 21941-909, Brazil

    Manuel Stadlbauer

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  1. Sébastien Gouëzel
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  2. Jérôme Rousseau
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  3. Manuel Stadlbauer
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All others together wrote the whole manuscript.

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Correspondence to Sébastien Gouëzel.

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Gouëzel, S., Rousseau, J. & Stadlbauer, M. Minimal distance between random orbits. Probab. Theory Relat. Fields (2024). https://doi.org/10.1007/s00440-024-01283-3

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  • DOI: https://doi.org/10.1007/s00440-024-01283-3

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