Hostname: page-component-784d4fb959-2wsk2 Total loading time: 0 Render date: 2025-07-16T07:58:17.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Module shifts and measure rigidity in linear cellular automata

Published online by Cambridge University Press:  16 September 2008

MARCUS PIVATO
MARCUS PIVATO*
Affiliation:
Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, Ontario, Canada K9J 7B8 (email: marcuspivato@trentu.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose ℛ is a finite commutative ring of prime characteristic, 𝒜 is a finite ℛ-module, 𝕄:=ℤD×ℕE, and Φ is an ℛ-linear cellular automaton on 𝒜𝕄. If μ is a Φ-invariant measure which is multiply σ-mixing in a certain way, then we show that μ must be the Haar measure on a coset of some submodule shift of 𝒜𝕄. Under certain conditions, this means that μ must be the uniform Bernoulli measure on 𝒜𝕄.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1945 - 1958
Copyright
Copyright © 2008 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.)

Article purchase

Temporarily unavailable

References

[1] Dummit, D. S. and Foote, R. M.. Abstract Algebra. Prentice Hall, Englewood Cliffs, NJ, 1991.Google Scholar
[2] de la Rue, T.. An introduction to joinings in ergodic theory. Discrete Contin. Dyn. Syst. 15(1) (2006), 121–142.CrossRefGoogle Scholar
[3] Einsiedler, M.. Isomorphism and measure rigidity for algebraic actions on zero-dimensional groups. Monatsh. Math. 144(1) (2005), 39–69.CrossRefGoogle Scholar
[4] Hedlund, G. A.. Endormorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[5] Host, B., Maass, A. and Martıńez, S.. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446.CrossRefGoogle Scholar
[6] Kitchens, B. P.. Expansive dynamics on zero-dimensional groups. Ergod. Th. & Dynam. Sys. 7(2) (1987), 249–261.CrossRefGoogle Scholar
[7] Kitchens, B.. Dynamics of Zd actions on Markov subgroups. Topics in Symbolic Dynamics and Applications (Temuco, 1997) (London Mathematical Society Lecture Note Seres, 279). Cambridge University Press, Cambridge, 2000, pp. 89–122.Google Scholar
[8] Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9(4) (1989), 691–735.CrossRefGoogle Scholar
[9] Kitchens, B. and Schmidt, K.. Markov subgroups of (Z/2Z)Z2. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). American Mathematical Society, Providence, RI, 1992, pp. 265–283.CrossRefGoogle Scholar
[10] Kitchens, B. and Schmidt, K.. Mixing sets and relative entropies for higher-dimensional Markov shifts. Ergod. Th. & Dynam. Sys. 13(4) (1993), 705–735.CrossRefGoogle Scholar
[11] Ledrappier, F.. Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B 287(7) (1978), A561–A563.Google Scholar
[12] Masser, D. W.. Mixing and linear equations over groups in positive characteristic. Israel J. Math. 142 (2004), 189–204.CrossRefGoogle Scholar
[13] Pivato, M.. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736.CrossRefGoogle Scholar
[14] Rudolph, D. J.. Fundamentals of Measurable Dynamics (Ergodic Theory on Lebesgue Spaces). Oxford University Press, New York, 1990.Google Scholar
[15] Sablik, M.. Measure rigidity for algebraic bipermutative cellular automata. Ergodic Th. & Dynam. Sys. 27(6) (2007), 1965–1990.CrossRefGoogle Scholar
[16] Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128). Birkhäuser, Basel, 1995.Google Scholar
[17] Schmidt, K.. Invariant measures for certain expansive Z2-actions. Israel J. Math. 90(1–3) (1995), 295–300.CrossRefGoogle Scholar
[18] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar