Mixing properties of one-dimensional

cellular automata

Author:
Rune Kleveland

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1755-1766

MSC (1991):
Primary 47A35, 22D25; Secondary 28D05, 46L05

MathSciNet review:
1363428

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Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of endomorphisms on the space of bi-infinite sequences over a finite set, and show that such a map is onto if and only if it is measure-preserving. A class of dynamical systems arising from these endomorphisms are strongly mixing, and some of them even -mixing. Some of these are isomorphic to the one-sided shift on in both the topological and measure-theoretical sense. Such dynamical systems can be associated to , the Cuntz-algebra of order , in a natural way.

**[Br-Jø-Pr]**Ola Bratteli, Palle E.T. Jørgensen and Geoffrey L. Price.*Endomorphisms of*. Proc. Symp. Pure Math. Am. math. soc. 1995. (ed. I.E. Segal.) To appear.**[Cuntz]**Joachim Cuntz,*Simple 𝐶*-algebras generated by isometries*, Comm. Math. Phys.**57**(1977), no. 2, 173–185. MR**0467330****[Hed]**G. A. Hedlund,*Endomorphisms and automorphisms of the shift dynamical system*, Math. Systems Theory**3**(1969), 320–375. MR**0259881****[Mat]**Kengo Matsumoto,*𝐶*-algebras associated with cellular automata*, Math. Scand.**75**(1994), no. 2, 195–216. MR**1319731****[Sh-Ro]**M. Shirvani and T. D. Rogers,*On ergodic one-dimensional cellular automata*, Comm. Math. Phys.**136**(1991), no. 3, 599–605. MR**1099698****[Wal]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**

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Additional Information

**Rune Kleveland**

Affiliation:
Department of Mathematics, University of Oslo, Box 1053, 0316 Oslo, Norway

Email:
runekl@math.uio.no

DOI:
https://doi.org/10.1090/S0002-9939-97-03708-8

Received by editor(s):
October 23, 1995

Received by editor(s) in revised form:
December 13, 1995

Communicated by:
Palle E. Jørgensen

Article copyright:
© Copyright 1997
American Mathematical Society