Mean dimension of Zk -actions

Yonatan Gutman, Elon Lindenstrauss, Masaki Tsukamoto

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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 languageEnglish
Pages (from-to)778-817
Number of pages40
JournalGeometric and Functional Analysis
Volume26
Issue number3
DOIs
Publication statusPublished - Jun 1 2016
Externally publishedYes

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Topological Entropy
Entropy
Inverse Limit
Topological Invariants
Infinite Dimensions
Compact Metric Space
Regular hexahedron
Open Problems
Isomorphic
Dynamical system
Orbit
Partition
Interpretation
Generalization

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

Cite this

Mean dimension of Zk -actions. / Gutman, Yonatan; Lindenstrauss, Elon; Tsukamoto, Masaki.

In: Geometric and Functional Analysis, Vol. 26, No. 3, 01.06.2016, p. 778-817.

Research output: Contribution to journalArticle

Gutman, Yonatan ; Lindenstrauss, Elon ; Tsukamoto, Masaki. / Mean dimension of Zk -actions. In: Geometric and Functional Analysis. 2016 ; Vol. 26, No. 3. pp. 778-817.
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