Geometric and Functional Analysis

, Volume 26, Issue 3, pp 778–817 | Cite as

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

  • Yonatan Gutman
  • Elon Lindenstrauss
  • Masaki Tsukamoto


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.

Mathematics Subject Classification

37B40 54F45 

Keywords and phrases

Mean dimension Topological entropy Metric mean dimension Voronoi tiling 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Yonatan Gutman
    • 1
  • Elon Lindenstrauss
    • 2
  • Masaki Tsukamoto
    • 3
    • 4
  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Einstein Institute of MathematicsHebrew UniversityJerusalemIsrael
  3. 3.Department of MathematicsKyoto UniversityKyotoJapan
  4. 4.Einstein Institute of MathematicsHebrew UniversityJerusalemIsrael

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