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Invariant measures for
$\mathscr {B}$-free systems revisited
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 10 / October 2024
- Published online by Cambridge University Press:
- 08 March 2024, pp. 2901-2932
- Print publication:
- October 2024
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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.
