Limiting measures for addition modulo a prime number cellular automata

Abstract

Linear cellular automata have many invariant measures in general. 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 - the most natural one. 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. In the two-state case, we characterize when iterates by addition modulo 2 cellular automata starting from a convex combination of strong mixing probability measures can converge. This also gives all invariant measures inside the class of those probability measures. We can obtain a similar result for iterates by addition modulo an odd prime number cellular automata starting from strong mixing probability measures.