## Abstract

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a Z^{k}-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 h_{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)When can we embed X into the Z^{k}-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 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.

Original language | English |
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Pages (from-to) | 778-817 |

Number of pages | 40 |

Journal | Geometric and Functional Analysis |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jun 2016 |

### Bibliographical note

Publisher Copyright:© 2016, Springer International Publishing.

## Keywords

- 37B40
- 54F45

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