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Computable Følner monotilings and a theorem of Brudno

Part of: Theory of computing Ergodic theory

Published online by Cambridge University Press:  13 November 2020

NIKITA MORIAKOV
NIKITA MORIAKOV*
Affiliation:
Department of Imaging, Radboud University Medical Center, Geert Grooteplein 10, 6525 GANijmegen, The Netherlands Department of Radiation Oncology, Netherlands Cancer Institute, Plesmanlaan 121, 1066 CXAmsterdam, The Netherlands
*
e-mail: [email protected]
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Abstract

A theorem of Brudno says that the Kolmogorov–Sinai entropy of an ergodic subshift over $\mathbb {N}$ equals the asymptotic Kolmogorov complexity of almost every word in the subshift. The purpose of this paper is to extend this result to subshifts over computable groups that admit computable regular symmetric Følner monotilings, which we introduce in this work. For every $d \in \mathbb {N}$ , the groups $\mathbb {Z}^d$ and $\mathsf{UT}_{d+1}(\mathbb {Z})$ admit computable regular symmetric Følner monotilings for which the required computing algorithms are provided.

Keywords

entropyKolmogorov complexitymeasure-preserving dynamical systems

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.)
Secondary: 37A15: General groups of measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3389 - 3416
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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