Invariant measures for -free systems revisited

Abstract

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