Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-11T01:15:36.828Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizing asymptotic randomization in abelian cellular automata

Part of: Topological dynamics Ergodic theory

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 Matemåticas, Universidad Andrés Bello, Chile email hellouin@lri.fr Centro de Modelamiento Matemåtico (CMM), Universidad de Chile, Chile
V. SALO
Affiliation:
Centro de Modelamiento MatemĂĄtico (CMM), Universidad de Chile, Chile
G. THEYSSIER
Affiliation:
Institut de Mathématiques de Marseille, Université Aix Marseille, CNRS, Centrale Marseille, France
Get access Rights & Permissions [Opens in a new window]

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

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

Boyer, L., Delacourt, M., Poupet, V., Sablik, M. and Theyssier, G.. 𝜇-limit sets of cellular automata from a computational complexity perspective. J. Comput. Syst. Sci. 81(8) (2015), 1623–1647.Google Scholar
Cattaneo, G., Finelli, M. and Margara, L.. Investigating topological chaos by elementary cellular automata dynamics. Theoret. Comput. Sci. 244(1) (2000), 219–241.Google Scholar
Ceccherini-Silberstein, T. and Coornaert, M.. Cellular Automata and Groups. Springer, Berlin, 2010.Google Scholar
Deitmar, A.. A First Course in Harmonic Analysis (Universitext). Springer, New York, 2002.Google Scholar
Delacourt, M. and Hellouin de Menibus, B.. Construction of 𝜇-limit sets of two-dimensional cellular automata. Proc. 32nd Int. Symp. on Theoretical Aspects of Computer Science, STACS 2015, March 4–7, 2015, Garching, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2015, pp. 262–274.Google Scholar
Delacourt, M., Poupet, V., Sablik, M. and Theyssier, G.. Directional dynamics along arbitrary curves in cellular automata. Theoret. Comput. Sci. 412(30) (2011), 3800–3821.Google Scholar
Ferrari, P. A., Maass, A., Martinez, S. and Ney, P.. Cesàro mean distribution of group automata starting from measures with summable decay. Ergod. Th. & Dynam. Sys. 20(6) (1999), 1657–1670.Google Scholar
Gajardo, A., Kari, J. and Moreira, A.. On time-symmetry in cellular automata. J. Comput. Syst. Sci. 78(4) (2012), 1115–1126.Google Scholar
Gajardo, A., Nesme, V. and Theyssier, G.. Pre-expansivity in cellular automata. Preprint, 2016, arXiv:1603.07215.Google Scholar
Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3 (1969), 320–375.Google Scholar
Hellouin de Menibus, B.. Asymptotic behaviour of cellular automata: computation and randomness. PhD Thesis, Aix-Marseille University, 2010.Google Scholar
Hellouin de Menibus, B. and Sablik, M.. Characterization of sets of limit measures of a cellular automaton iterated on a random configuration. Ergod. Th. & Dynam. Sys. (2016), 1–50, 001.Google Scholar
Host, B., Maass, A. and Martínez, S.. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446.Google Scholar
Kari, J. and Taati, S.. Statistical mechanics of surjective cellular automata. J. Stat. Phys. 160(5) (2015), 1198–1243.Google Scholar
LĂ©vy, P.. Sur la dĂ©termination des lois de probabilitĂ© par leurs fonctions caractĂ©ristiques. C. R. Acad. Sci. Paris 175 (1922), 854–856.Google Scholar
Lind, D. A.. Applications of ergodic theory and sofic systems to cellular automata. Physica D 10(1–2) (1984), 36–44.Google Scholar
Maass, A. and Martínez, S.. Time Averages for Some Classes of Expansive One-dimensional Cellular Automata. Springer, Dordrecht, 1999, pp. 37–54.Google Scholar
Maass, A., Martínez, S., Pivato, M. and Yassawi, R.. Asymptotic randomization of subgroup shifts by linear cellular automata. Ergod. Th. & Dynam. Sys. 26(8) (2006), 1203–1224.Google Scholar
Miyamoto, M.. An equilibrium state for a one-dimensional life game. J. Math. Kyoto Univ. 19(3) (1979), 525–540.Google Scholar
Parthasarathy, K. R., Ranga Rao, R. and Varadhan, S. R. S.. Probability distributions on locally compact abelian groups. Illinois J. Math. 7(2) (1963), 337–369.Google Scholar
Pivato, M.. Ergodic theory of cellular automata. Encyclopedia of Complexity and Systems Science. Ed. Meyers, R. A.. Springer, New York, NY, 2009, pp. 2980–3015.Google Scholar
Pivato, M. and Yassawi, R.. Limit measures for affine cellular automata. Ergod. Th. & Dynam. Sys. 22 (2002), 1269–1287.Google Scholar
Pivato, M. and Yassawi, R.. Limit measures for affine cellular automata II. Ergod. Th. & Dynam. Sys. 30 (2003), 1–20.Google Scholar
Pivato, M. and Yassawi, R.. Asymptotic randomization of sofic shifts by linear cellular automata. Ergod. Th. & Dynam. Sys. 26(4) (2006), 1177–1201.Google Scholar
Sablik, M.. Directional dynamics for cellular automata: a sensitivity to initial condition approach. Theoret. Comput. Sci. 400(1–3) (2008), 1–18.Google Scholar
Salo, V.. Subshifts with sparse projective subdynamics. Preprint, 2016, arXiv:1605.09623.Google Scholar
Salo, V. and TörmĂ€, I.. Commutators of bipermutive and affine cellular automata. Proc. Int. Workshop on Cellular Automata and Discrete Complex Systems (Gießen, Germany, 17–19 September 2013). Springer, Berlin, 2013, pp. 155–170.Google Scholar
Sobottka, M.. Right-permutive cellular automata on topological Markov chains. Discrete Contin. Dyn. Syst. A 20(4) (2008), 1095–1109.Google Scholar
Taati, S.. Statistical equilibrium in deterministic cellular automata. Probabilistic Cellular Automata: Theory, Applications and Future Perspectives. Eds. Louis, P.-Y. and Nardi, F. R.. Springer International Publishing, Berlin, 2018, pp. 145–164.Google Scholar