Generalized heredity in B-free systems - Keller, Gerhard (Stochastics and Dynamics, 2021)
Let B N{1 be a primitive set, FB := ZbZ, := 1FB 0, 1Z, and denote by X the orbit closure of under the shift. We complement results on heredity of X from [Dymek et al., B-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425 5489] in two directions: In the proximal case we prove that a certain subshift X X, which coincides with X when B is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set B has light tails, as in [Dymek et al., B-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425 5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that X is always hereditary above its unique minimal (Toeplitz) subsystem . Finally, we characterize this Toeplitz subsystem as being a set X, where = 1FB for a set B that can be derived from B, and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of B-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690 2734] are heavily used.
CRIS-Team Scopus Importer:2021-04-19
Keywords: B-free dynamics; heredity; sets of multiples