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Mean dimension of \({\mathbb{Z}^k}\)-actions

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Abstract

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

  1. (1)

    When is X isomorphic to the inverse limit of finite entropy systems?

  2. (2)

    Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?

  3. (3)

    When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).

A key ingredient is a new method to continuously partition every orbit into good pieces.

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Author notes

  1. Masaki Tsukamoto

    Present address: Einstein Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland

    Yonatan Gutman

  2. Einstein Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Elon Lindenstrauss

  3. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Tsukamoto

Authors
  1. Yonatan Gutman
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  2. Elon Lindenstrauss
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  3. Masaki Tsukamoto
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Corresponding author

Correspondence to Masaki Tsukamoto.

Additional information

Y.G. was partially supported by the Marie Curie grant PCIG12-GA-2012-334564 and by the National Science Center (Poland) grant 2013/08/A/ST1/00275. E.L. acknowledges the support of ERC AdG Grant 267259. M.T. was supported by Grant-in-Aid for Young Scientists (B) 25870334 from JSPS.

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Gutman, Y., Lindenstrauss, E. & Tsukamoto, M. Mean dimension of \({\mathbb{Z}^k}\)-actions. Geom. Funct. Anal. 26, 778–817 (2016). https://doi.org/10.1007/s00039-016-0372-9

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  • DOI: https://doi.org/10.1007/s00039-016-0372-9

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