Limit measures for affine cellular automata

Abstract

Let \mathbb{M} be a monoid (e.g. \mathbb{N}, \mathbb{Z}, or \mathbb{M}^D), and \mathcal{A} an abelian group. \mathcal{A}^\mathbb{M} is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrak{F}:\mathcal{A}^\mathbb{M}\longrightarrow\mathcal{A}^\mathbb{M} that commutes with all shift maps. Let \mu be a (possibly non-stationary) probability measure on \mathcal{A}^\mathbb{M}; we develop sufficient conditions on \mu and \mathfrak{F} so that the sequence \{\mathfrak{F}^N\mu\}_{N=1}^\infty weak* converges to the Haar measure on \mathcal{A}^\mathbb{M} in density (and thus, in Cesàro average as well). As an application, we show that, if \mathcal{A}=\mathbb{Z}_{/p} (p prime), \mathfrak{F} is any ‘non-trivial’ LCA on \mathcal{A}^{(\mathbb{Z}^D)}, and \mu belongs to a broad class of measures (including most Bernoulli measures (for D \geq 1) and ‘fully supported’ N-step Markov measures (when D=1)), then \mathfrak{F}^N\mu weak* converges to the Haar measure in density.