Then f is weakly preperiodic if and only if it is preperiodic.Īcknowledgments. Let G be a countable group, X AG be a subshift, and f W X ! XĪ cellular automaton. Theorem 1.6, by applying it to the semigroup N.Ĭorollary 1.7. To all subshifts on the groups Zd, and this proof works on all countable groups,īut the proof does not apply to preperiodicity. In, the result about nilpotency is generalized In, it is shown that if X is a transitive Z-subshift, andį W X ! X is a cellular automaton which is weakly preperiodic (weakly nilpotent), x/ D y for some n, and nilpotent if n is uniform. Similarly f is weakly nilpotent if for some uniform configuration y 2 X and all ForĢ C RW s 2 0 0 j 0, and preperiodic if f n D f nCp for some n 0 p > 0.
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While Kaul’s result can be generalizedĬonsiderably (see ), some topological assumptions about the space are crucial.įor instance, consider the following example, where the assumption “X is aĮxample 1.2 (locally connected compact tree with pointwise periodic action). Transformation group is always compact when the transformation group acts on aĬonnected manifold without boundary. Generalizing a result of Montgomery, Kaul showed that a pointwise periodic Without extra assumptions, it is completely trivial that the answer is false. Is a pointwise periodic transformation group finite (compact)? Įquivalently, all G -orbits are finite (compact). X is pointwise periodic if the stabilizer of every x 2 X is of finite index in G. įollowing Kaul, a discrete (topological) group G of transformations of set ħ Concluding remarks and further directions. ĥ Pointwise periodic automorphism groups of expansive actions. ģ Non-expansive horoballs for actions of finitely generated groups. Expansiveness, tilings, subshifts.ġ Introduction. Mathematics Subject Classification (2010). finite, and related results for tilings of Euclidean space.
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WeĪlso prove that a subshift over any finitely generated group that consists of finite orbits is Pointwise periodic finitely generated group of cellular automata is necessarily finite. Groups and semigroups of transformations. We study implications of expansiveness and pointwise periodicity for certain