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

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 that there exists an almost diam-mean equicontinuous CA that is not almost equicontinuous (and hence not almost locally periodic). Previously, we constructed a CA that is almost mean equicontinuous [L.D.I.S. Banos and F. Garcia-Ramos, Mean equicontinuity and mean sensitivity on cellular automata, Ergodic Theory Dynam. Systems 41 (12) (2021), pp. 3704-3721] but not almost diam-mean equicontinuous [L.D.I.S. Banos and F. Garcia-Ramos, Diameter mean equicontinuity and cellular automata, Proceedings of the 27th International Workshop on Cellular Automata and Discrete Complex Systems, arXiv:2106.09641, 2021].