Mixing properties of one-dimensional cellular automata

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.