TY - JOUR

T1 - Mean dimension of Zk -actions

AU - Gutman, Yonatan

AU - Lindenstrauss, Elon

AU - Tsukamoto, Masaki

N1 - Publisher Copyright:
© 2016, Springer International Publishing.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00039-016-0372-9

DO - 10.1007/s00039-016-0372-9

M3 - Article

AN - SCOPUS:84976407186

VL - 26

SP - 778

EP - 817

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 3

ER -