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Topological entropy zero and asymptotic pairs

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Abstract

Let T be a continuous map on a compact metric space (X, d). A pair of distinct points x, y ∈ X is asymptotic if lim n→∞ d(T n x, T n y) = 0. We prove the following four conditions to be equivalent: 1. h top(T) = 0; 2. (X, T) has a (topological) extension (Y,S) which has no asymptotic pairs; 3. (X, T) has a topological extension (Y ′, S′) via a factor map that collapses all asymptotic pairs; 4. (X, T) has a symbolic extension (i.e., with (Y ′, S′) being a subshift) via a map that collapses asymptotic pairs. The maximal factors (of a given system (X, T)) corresponding to the above properties do not need to coincide.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    T. Downarowicz

  2. Institut des Sciences de l’Ingénieur de Toulon et du Var, Avenue G. Pompidou, B.P. 56, 83162, La Valette du Var Cedex, France

    Y. Lacroix

Authors
  1. T. Downarowicz
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  2. Y. Lacroix
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Correspondence to T. Downarowicz.

Additional information

Research of the first author is supported from resources for science in years 2009–2012 as research project (grant MENII N N201 394537, Poland)

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Downarowicz, T., Lacroix, Y. Topological entropy zero and asymptotic pairs. Isr. J. Math. 189, 323–336 (2012). https://doi.org/10.1007/s11856-011-0174-6

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  • DOI: https://doi.org/10.1007/s11856-011-0174-6

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