  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.
