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

TL;DR

This paper extends the understanding of invariant measures and entropy for $$-free systems, confirming a conjecture and providing new insights under the assumption that the Toeplitz sequence is regular.

Contribution

It proves that many measure and entropy results hold for the original $$-free subshift, not just its hereditary closure, and confirms Keller's conjecture on measure description.

Findings

Invariant measures for $X_$ are characterized similarly to the hereditary closure.

The conjecture of Keller on measure description is affirmed.

Regularity of the Toeplitz sequence is key to the analysis.

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.