Asymptotic Randomization of Sofic Shifts by Linear Cellular Automata

Abstract

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).