  Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is
then a compact abelian group; a `linear cellular automaton' (LCA) is a
topological group endomorphism \Phi:A^M --> A^M that commutes with all shift
maps.
  Suppose \mu is a probability measure on A^M whose support is a subshift of
finite type or sofic shift. We provide sufficient conditions (on \Phi and \mu)
under which \Phi `asymptotically randomizes' \mu, meaning that wk*lim_{J\ni j
--> oo} \Phi^j \mu = \eta, where \eta is the Haar measure on A^M, and J has
Cesaro density 1. In the case when \Phi=1+\sigma, we provide a condition on \mu
that is both necessary and sufficient. We then use this to construct an example
of a zero-entropy measure which is asymptotically randomized by 1+\sigma (all
previously known examples had positive entropy).
