Local non-periodic order and diam-mean equicontinuity on cellular automata

TL;DR

This paper explores the concept of diam-mean equicontinuity in cellular automata, demonstrating the existence of automata with specific non-periodic order properties and challenging existing dichotomies in dynamical systems.

Contribution

It introduces a novel example of a cellular automaton that is almost diam-mean equicontinuous but not almost equicontinuous, and shows that Kurka's dichotomy does not extend to diam-mean sensitivity and equicontinuity.

Findings

Existence of an almost diam-mean equicontinuous CA that is not almost equicontinuous

Counterexample to Kurka's dichotomy in diam-mean sensitivity and equicontinuity

Demonstration of local skew product construction in cellular automata

Abstract

Diam-mean equicontinuity is a dynamical property that has been of use in the study of non-periodic order. Using some type of "local" skew product between a shift and an odometer looking cellular automaton (CA), we will show there exists an almost diam-mean equicontinuous CA that is not almost equicontinuous, (and hence not almost locally periodic). As an application we show that Kurka's dichotomy does not hold for diam-mean versions of sensitivity and equicontinuity.