2017/04/13 by Gerhard Keller, Keller, Gerhard
Computer Science · Mathematics · #Cellular Automata and Applications #Mathematical Dynamics and Fractals #math.DS #msc:37A35 #msc:37A45 #msc:37B05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1704.04079
The statement and the proof of Lemma 4 (current version) were corrected. In view of [arxiv:1802.08309], stronger corrolaries to the main result could be formulated
arxiv created 2019/11/15 · arxiv updated 2019/11/18
Let \mathcal B⊆\mathbb N be a primitive set. We complement results on heredity of the \mathcal B-free subshift Xη from [arxiv:1509.08010] in two directions: In the proximal case we prove that a subshift Xφ, which micht be slightly larger than the subshift Xη, is always hereditary. (There is no need to assume that the set \mathcal B is taut or even has light tails, but if \mathcal B is taut, then Xφ=Xη.) We also generalize the the concept of heredity to the non-proximal (and hence non-hereditary) case by proving that Xφ is always "hereditary away from its unique minimal subsystem" (which is always Toeplitz). Finally we characterize regularity of this Toeplitz subsystem equivalently by the condition mH(int(W))=0, where W ("the window") is a subset of a compact abelian group H canonically associated with the set \mathcal B, and mH denotes Haar measure on H. Throughout, results from [arxiv:1702.02375] are heavily used.