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Classification of backward filtrations and factor filtrations: examples from cellular automata

Part of: Topological dynamics Markov processes Ergodic theory

Published online by Cambridge University Press:  21 July 2021

PAUL LANTHIER  and
THIERRY DE LA RUE Open the ORCID record for THIERRY DE LA RUE [Opens in a new window]
PAUL LANTHIER
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, F-76801 Saint Étienne du Rouvray, France (e-mail: planthier76@outlook.fr)
THIERRY DE LA RUE*
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, F-76801 Saint Étienne du Rouvray, France (e-mail: planthier76@outlook.fr)
*
e-mail: Thierry.de-la-Rue@univ-rouen.fr
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Abstract

We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik’s theory, but we also study them from another point of view that takes into account the measure-preserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.

Keywords

filtrationsI-cosinessmeasure-theoretic factorscellular automata

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37B15: Cellular automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2890 - 2922
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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