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Characterizing asymptotic randomization in abelian cellular automata

Part of: Ergodic theory Topological dynamics

Published online by Cambridge University Press:  25 September 2018

B. HELLOUIN DE MENIBUS Open the ORCID record for B. HELLOUIN DE MENIBUS [Opens in a new window] ,
V. SALO Open the ORCID record for V. SALO [Opens in a new window]  and
G. THEYSSIER
B. HELLOUIN DE MENIBUS
Affiliation:
Departamento de Matemáticas, Universidad Andrés Bello, Chile email hellouin@lri.fr Centro de Modelamiento Matemático (CMM), Universidad de Chile, Chile
V. SALO
Affiliation:
Centro de Modelamiento Matemático (CMM), Universidad de Chile, Chile
G. THEYSSIER
Affiliation:
Institut de Mathématiques de Marseille, Université Aix Marseille, CNRS, Centrale Marseille, France
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Abstract

Abelian cellular automata (CAs) are CAs which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images have weak*-convergence towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i.e., randomization for a wide class of initial measures (under some mixing hypotheses). First, we prove that an abelian CA randomizes in Cesàro mean if and only if it has no soliton, i.e., a non-zero finite configuration whose time evolution remains bounded in space. This characterization generalizes previously known sufficient conditions for abelian CAs with scalar or commuting coefficients. Second, we exhibit examples of strong randomizers, i.e., abelian CAs randomizing in simple convergence; this is the first proof of this behaviour to our knowledge. We show, however, that no CA with commuting coefficients can be strongly randomizing. Finally, we show that some abelian CAs achieve partial randomization without being randomizing: the distribution of short finite words tends to the uniform distribution up to some threshold, but this convergence fails for larger words. Again this phenomenon cannot happen for abelian CAs with commuting coefficients.

Keywords

cellular automataclassical ergodic theorysymbolic dynamics

MSC classification

Primary: 37B15: Cellular automata
Secondary: 37A60: Dynamical systems in statistical mechanics 37A05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 923 - 952
Copyright
© Cambridge University Press, 2018

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