Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T18:25:11.029Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Topological dynamics 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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bramson, M. and Neuhauser, C.. Survival of one-dimensional cellular automata under random perturbations. Ann. Probab. 22(1) (1994), 244–263.CrossRefGoogle Scholar
Bressaud, X., Maass, A., Martinez, S. and San Martin, J.. Stationary processes whose filtrations are standard. Ann. Probab. 34(4) (2006), 1589–1600.CrossRefGoogle Scholar
Ceillier, G. and Leuridan, C.. Sufficient conditions for the filtration of a stationary processes to be standard. Probab. Theory Related Fields 167(3–4) (2017), 979–999.CrossRefGoogle Scholar
de la Rue, T.. An introduction to joinings in ergodic theory . Discrete Contin. Dyn. Syst. 15(1) (2006), 121–142.CrossRefGoogle Scholar
Dubins, L., Feldman, J., Smorodinsky, M. and Tsirelson, B.. Decreasing sequences of $\sigma$ -fields and a measure change for Brownian motion. I. Ann. Probab. 24(2) (1996), 882–904.CrossRefGoogle Scholar
Émery, M. and Schachermayer, W.. On Vershik’s standardness criterion and Tsirelson’s notion of cosiness . Séminaire de Probabilités XXXV. Ed. J. Azéma, M. Émery, M. Ledoux and M.Yor. Springer, Berlin, 2001, pp. 265–305.CrossRefGoogle Scholar
Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation . Math. Syst. Theory 1 (1967), 1–49.CrossRefGoogle Scholar
Hoffman, C.. A zero entropy $\mathrm{T}$ such that the $\left[\mathrm{T},\mathrm{Id}\right]$ endomorphism is nonstandard. Proc. Am. Math. Soc. 128(1) (2000), 183–188.CrossRefGoogle Scholar
Hoffman, C. and Rudolph, D.. A dyadic endomorphism which is Bernoulli but not standard. Israel J. Math. 130 (2002), 365–379.CrossRefGoogle Scholar
Lanthier, P.. Aspects ergodiques et algébriques des automates cellulaires. PhD Thesis, Université de Rouen Normandie, 2020.Google Scholar
Laurent, S.. On standardness and I-cosiness . Séminaire de Probabilités XLIII, Poitiers, France, Juin 2009. Ed. C. Donati-Martin, A. Lejay and A.Rouault. Springer, Berlin, 2011, pp. 127–186.Google Scholar
Liggett, T. M.. Survival of discrete time growth models, with applications to oriented percolation. Ann. Appl. Probab. 5(3) (1995), 613–636.CrossRefGoogle Scholar
Marcovici, I.. Ergodicity of noisy cellular automata: the coupling method and beyond . Pursuit of the Universal. Proc. 12th Conf. on Computability in Europe, CiE 2016 (Paris, France, June 27–July 1, 2016). Ed. Beckmann, A., Bienvenu, L. and Jonoska, N.. Springer, Cham, 2016, pp. 153–163.Google Scholar
Smorodinsky, M.. Processes with no standard extension . Israel J. Math. 107 (1998), 327–331.CrossRefGoogle Scholar
Tsirelson, B.. Triple points: from non-Brownian filtrations to harmonic measures . Geom. Funct. Anal. 7(6) (1997), 1096–1142.CrossRefGoogle Scholar
Vershik, A. M.. Decreasing sequences of measurable partitions and their applications. Sov. Math. Dokl. 11 (1970), 1007–1011.Google Scholar
Vershik, A. M.. The theory of decreasing sequences of measurable partitions . St. Petersburg Math. J. 6(4) (1994), 1–68.Google Scholar