Invariant measures for -free systems revisited

For B ⊆ N , the B -free subshift Xη is the orbit closure of the characteristic function of the set of B -free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of Xη , have their analogues for Xη as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in B -free systems. Stoch. Dyn. 21 (3) (2021), Paper No. 2140008]. A central assumption in our work is that η * (the Toeplitz sequence that generates the unique minimal component of Xη ) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in Xη from above and below.