### 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 |
---|---|

Pages (from-to) | 778-817 |

Number of pages | 40 |

Journal | Geometric and Functional Analysis |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 1 2016 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

### Cite this

^{k}-actions.

*Geometric and Functional Analysis*,

*26*(3), 778-817. https://doi.org/10.1007/s00039-016-0372-9

**Mean dimension of Z ^{k} -actions.** / Gutman, Yonatan; Lindenstrauss, Elon; Tsukamoto, Masaki.

Research output: Contribution to journal › Article

^{k}-actions',

*Geometric and Functional Analysis*, vol. 26, no. 3, pp. 778-817. https://doi.org/10.1007/s00039-016-0372-9

^{k}-actions. Geometric and Functional Analysis. 2016 Jun 1;26(3):778-817. https://doi.org/10.1007/s00039-016-0372-9

}

TY - JOUR

T1 - Mean dimension of Zk -actions

AU - Gutman, Yonatan

AU - Lindenstrauss, Elon

AU - Tsukamoto, Masaki

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.

UR - http://www.scopus.com/inward/record.url?scp=84976407186&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976407186&partnerID=8YFLogxK

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 -