A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation Phi : A(Z)-> A(Z) determined by a local rule phi : A({0,1}) -> A so that, for any a is an element of A(Z) and any z is an element of Z, Phi(a)(z) = phi(a(z), a(z) + 1). We say that Phi is bipermutative if, for any choice of a is an element of A, the map A there exists b -> phi(a, b) is an element of A is bijective, and also, for any choice of b is an element of A, the map A there exists a -> phi(a, b) is an element of A is bijective.We characterize the invariant measures of bipermutative RNNCA. First we introduce the equivalent notion of a quasigroup CA. Then we characterize Phi-invariant measures when A is a (nonabelian) group, and phi(a, b) = a circle b. Then we show that, if Phi is any bipermutative RNNCA, and mu is Phi-invariant, then Phi must be mu-almost everywhere K-to-1, for some constant K. We then characterize invariant measures when A(Z) is a group shift and Phi is an endomorphic CA.
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