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

Aurelia Dymek, Joanna Kułaga-Przymus, Daniel Sell

Abstract

For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_η $ 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_η $, have their analogues for $ X_η $ 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 $ η^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_η $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_η $ from above and below.

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

Abstract

For , the -free subshift is the orbit closure of the characteristic function of the set of -free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of , have their analogues for as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in -free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that (the Toeplitz sequence that generates the unique minimal component of ) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in from above and below.
Paper Structure (34 sections, 22 theorems, 129 equations)

This paper contains 34 sections, 22 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and main objects
  4. Dynamics
  5. Toeplitz systems
  6. B-free systems
  7. Subshifts
  8. Basic algebraic objects
  9. Group homomorphisms
  10. More subshifts
  11. Dynamical diagrams
  12. Summary
  13. Main results: invariant measures
  14. Main results: tautness and combinatorics
  15. Main results: entropy
  16. ...and 19 more sections

Key Result

Lemma 1.2

Let $h\in H$. Then $\Gamma_{H,H^*}(h)_{b^*}=h_b\bmod b^*$ for any $b\in{\mathscr{B}}$ and any $b^*\in{\mathscr{B}}^*$ such that $b^*\mid b$. In particular, $\Gamma_{H,H^*}(h)_b=h_b$ for any $b\in{\mathscr{B}}\cap{\mathscr{B}}^*$.

Theorems & Definitions (51)

  • Remark 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • ...and 41 more