arXiv:math/0108084 Abstract

arXiv:math/0108084 [math.DS]Abstract References Reviews Resources

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

Published 2001-08-12, updated 2002-08-28Version 3

If M is a monoid (e.g. the lattice Z^D), and G is a finite (nonabelian) group, then G^M is a compact group; a `multiplicative cellular automaton' (MCA) is a continuous transformation F:G^M-->G^M 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 G^M, and show that MCA on G^M 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.

Comments: LaTeX2E Format, 20 pages, 2 tables. To appear in Journal of Statistical Physics. REVISION 1: extensive notational improvement. Some results extended or improved; minor errors corrected. REVISION 2: Title changed (Old title was ``Nonabelian Multiplicative Cellular Automata''). Some proofs/examples suppressed for brevity. Some sections reorganized. Minor errors corrected

Categories: math.DS

Subjects: 37B15, 68Q80

Keywords: multiplicative cellular automaton, nilpotent groups, asymptotics, natural structure theory, initial measures

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