Prevalence of Odometers in Cellular Automata

We consider left permutive cellular automata Φ with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points x, which is large in another sense too, such that Φ : cl{Φn(x) : n ≥ 0} → cl{Φn(x) : n ≥ 0} is topologically conjugate to an odometer, the “+1” map on the countable product of finite cyclic groups. We identify the odometer in several cases.