## Abstract

The commonly used topological entropy htop(U) of the multidimensional shift space U is the rectangular spatial entropy h_{r}(U) which is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space ℤ^{d}, d ≥ 2. This work studies spatial entropy h_{Ω}(U) of shift space U on general expanding system _{Ω} = {(n)}_{n=1}^{∞} where _{Ω} (n) is increasing finite sublattices and expands to ℤ^{d}. _{Ω} is called genuinely d-dimensional if _{Ω} (n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that h_{r}(U) is the supremum of h_{Ω} (U) for all genuinely d-dimensional _{Ω}. Furthermore, when is genuinely d-dimensional and satisfies certain conditions, then h_{Ω} (U) = h_{r}(U). On the contrary, when _{Ω} (n) contains a lower-dimensional part which is comparable to its d-dimensional part, then h_{r}(U) < h_{Ω} (U) for some U. Therefore, h_{r}(U) is appropriate to be the d-dimensional spatial entropy.

Original language | English |
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Pages (from-to) | 3705-3718 |

Number of pages | 14 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 36 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jul 2016 |

## Keywords

- Block gluing
- Shift space
- Spatial entropy
- Symbolic dynamical system
- Topological entropy