Finally, we show that some abelian CA achieve partial randomization without being randomizing: the distribution of short finite words tends to the uniform distribution up to some threshold, but this convergence fails for larger words. Note that no requirement, even on halting, is made when they are distinct and conjugate. We show however that no CA with commuting coefficients can be strongly randomizing. There is no algorithm that given two context-free languages, says 'same' if they define the same subshift, and says 'non-conjugate' if they are not conjugate. A strategy would be to consider one by one factor maps of increasing radius. Thus, it is natural to try to produce (to compute) the finite list of all such factors of a given minimal substitutive subshift (X, S) (up to isomorphism). abelian CA randomizing in simple convergence this is the first proof of this behaviour to our knowledge. From Dur00, Dur13b, minimal morphic subshifts have finitely many aperiodic subshift factors. Second, we exhibit examples of strong randomizers, i.e. This characterization generalizes previously known sufficient conditions for abelian CA with scalar or commuting coefficients. a finite configuration whose time evolution remains bounded in space. For actions of the group $\mathbbro mean if and only if it has no soliton, i.e. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is S-adic with Card(S) 3 27. Our main result in particular shows that the class of nil-rigid groups includes all residually finite solvable groups and all groups of polynomial growth. We show that this property of nil-rigidity holds for a large class of groups and actions that satisfy a natural specification-like property and have dense homoclinic points. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$.
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