Local non-periodic order and diam-mean equicontinuity on 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.