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Hereditary subshifts whose measure of maximal entropy has no Gibbs property

Joanna Kułaga-Przymus, Michał Lemańczyk

Abstract

We show that the measure of maximal entropy for the hereditary closure of a $\mathscr{B}$-free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that $\mathscr{B}$ is taut whenever the corresponding Mirsky measure $ν_η$ has full support. This is the converse theorem to a recent result of Keller.

Hereditary subshifts whose measure of maximal entropy has no Gibbs property

Abstract

We show that the measure of maximal entropy for the hereditary closure of a -free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that is taut whenever the corresponding Mirsky measure has full support. This is the converse theorem to a recent result of Keller.

Paper Structure

This paper contains 17 sections, 17 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Hereditary subshifts
  3. Convolution measures
  4. Entropy
  5. Gibbs property
  6. $\mathscr{B}$-free systems
  7. Zoo of densities
  8. Entropy vs density
  9. Results
  10. Absence of Gibbs property
  11. Mirsky measure
  12. Topological support
  13. Atomic Mirsky measure
  14. Atoms
  15. Gibbs property vs entropy
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X,T)$ be a topological dynamical system. Let $x\in X$ and let $A\subset X$ be a clopen set. Then

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Theorem 2.6 in Bergelson_2008
  • proof : Proof of Theorem \ref{['prop11']}
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Example 1.1
  • Example 1.2
  • Lemma 1.6
  • proof
  • ...and 34 more