About a low complexity class of Cellular Automata

TL;DR

This paper extends the concept of m-equicontinuous cellular automata to all probability measures, demonstrating entropy nullity, measure convergence, and density of periodic points, thus revealing a low complexity class of cellular automata.

Contribution

It generalizes m-equicontinuous cellular automata beyond Bernoulli measures, showing entropy properties and measure convergence for invariant measures.

Findings

Entropy is zero for invariant measures under these automata.

Image measures converge in Cesaro mean to an invariant measure mc.

Periodic points are dense in the support of mc.

Abstract

Extending to all probability measures the notion of m-equicontinuous cellular automata introduced for Bernoulli measures by Gilman, we show that the entropy is null if m is an invariant measure and that the sequence of image measures of a shift ergodic measure by iterations of such automata converges in Cesaro mean to an invariant measure mc. Moreover this cellular automaton is still mc-equicontinuous and the set of periodic points is dense in the topological support of the measure mc. The last property is also true when m is invariant and shift ergodic.