Invariant Measures for Bipermutative Cellular Automata

A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation Φ : A−→A determined by a local rule φ : A{0,1}−→A so that, for any a ∈ A and any z ∈ Z, Φ(a)z = φ(az, az+1). We say that Φ is bipermutative if, for any choice of a ∈ A, the map A ∋ b 7→ φ(a, b) ∈ A is bijective, and also, for any choice of b ∈ A, the map A ∋ a 7→ φ(a, b) ∈ A is bijective. We characterize the invariant measures of bipermutative RNNCA. First we introduce the equivalent notion of a quasigroup CA, to expedite the construction of examples. Then we characterize Φ-invariant measures when A is a (nonabelian) group, and φ(a, b) = a · b. Then we show that, if Φ is any bipermutative RNNCA, and μ is Φ-invariant, then Φ must be μ-almost everywhere K-to-1, for some constant K. We use this to characterize invariant measures when A is a group shift and Φ is an endomorphic CA. MSC: Primary: 37B15; Secondary: 37A50