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arXiv · 2307.02134

Invariant measures for $\mathscr{B}$-free systems revisited

Aurelia Dymek·Joanna Kułaga-Przymus·Daniel Sell
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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 ([Keller, G. 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.

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Aurelia Dymek, Joanna Kułaga-Przymus, Daniel Sell. 2023-07-05. Invariant measures for $\mathscr{B}$-free systems revisited. https://arxiv.org/abs/2307.02134

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