On Limiting Measures for a Class of One-Dimensional Linear Cellular Automata

Abstract

Linear cellular automata have many invariant measures in general, but the most natural one is the uniform Bernoulli product measure. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. We characterize when iterates by addition modulo a prime number cellular automata starting from a strong mixing probability measure with full support can converge. This also gives all invariant measures inside the class of those probability measures. In the two-state case, we also obtain a necessary and sufficient condition that a convex combination of strong mixing probability measures is invariant under addition modulo 2 cellular automata. Those results improve previous ones obtained by Marcovici and Miyamoto.