Mixing properties of one-dimensional cellular automata

Kleveland, Rune

doi:10.1090/S0002-9939-97-03708-8

Mixing properties of one-dimensional cellular automata
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Proc. Amer. Math. Soc. 125 (1997), 1755-1766 Request permission

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 $m$-mixing. Some of these are isomorphic to the one-sided shift on $\mathbb {Z}_n$ in both the topological and measure-theoretical sense. Such dynamical systems can be associated to $\mathcal {O}_n$, the Cuntz-algebra of order $n$, in a natural way.

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