Limit Measures for Affine Cellular Automata
Abstract
Let M be a monoid (e.g. the lattice Z^D), and A an abelian group. A^M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M > A^M that commutes with all shift maps. Let mu be a (possibly nonstationary) probability measure on A^M; we develop sufficient conditions on mu and F so that the sequence {F^N mu} (N=1,2,3,...) weak*converges to the Haar measure on A^M, in density (and thus, in Cesaro average as well). As an application, we show: if A=Z/p (p prime), F is any ``nontrivial'' LCA on A^{(Z^D)}, and mu belongs to a broad class of measures (including most Bernoulli measures (for D >= 1) and ``fully supported'' Nstep Markov measures (when D=1), then F^N mu weak*converges to Haar measure in density.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2001
 arXiv:
 arXiv:math/0108082
 Bibcode:
 2001math......8082P
 Keywords:

 Mathematics  Dynamical Systems;
 37B15;
 68Q80
 EPrint:
 LaTeX2E Format, revised version contains minor corrections