Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics
Marcus Pivato
Abstract
If M is a monoid (e.g. the lattice ZD), and G is a finite (nonabelian) group, then GM is a compact group; a `multiplicative cellular automaton' (MCA) is a continuous transformation F:GM-->GM which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of G. We characterize when MCA are group endomorphisms of GM, and show that MCA on GM inherit a natural structure theory from the structure of G. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.
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