Mean dimension of Zk -actions

Yonatan Gutman, Elon Lindenstrauss, Masaki Tsukamoto*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a Zk-action on a compact metric space X, we study the following three problems closely related to mean dimension.(1)When is X isomorphic to the inverse limit of finite entropy systems?(2)Suppose the topological entropy htop(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)When can we embed X into the Zk-shift on the infinite dimensional cube ([0,1]D)Zk? These were investigated for Z-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to Zk 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.

Original languageAmerican English
Pages (from-to)778-817
Number of pages40
JournalGeometric and Functional Analysis
Issue number3
StatePublished - 1 Jun 2016

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing.


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