2003/06/12 by Marcus Pivato, Pivato, Marcus
Mathematics · #37A50 (Secondary) #37B15 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37A50 #msc:37B15
paper · pdf · doi:10.48550/arxiv.math/0306211
20 pages, 1 figure, 1 table
arxiv created 2003/06/12 · arxiv updated 2009/11/30
A `right-sided, nearest neighbour cellular automaton' (RNNCA) is a continuous transformation F:AZ-->AZ determined by a local rule f:A0,1-->A so that, for any a in AZ and any z in Z, F(a)z = f(az,az+1) . We say that F is `bipermutative' if, for any choice of a in A, the map g:A-->A defined by g(b) = f(a,b) is bijective, and also, for any choice of b in A, the map h:A-->A defined by h(a)=f(a,b) 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 F-invariant measures when A is a (nonabelian) group, and f(a,b) = a*b. Then we show that, if F is any bipermutative RNNCA, and mu is F-invariant, then F must be mu-almost everywhere K-to-1, for some constant K . We use this to characterize invariant measures when AZ is a `group shift' and F is an `endomorphic CA'.