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Bohr chaoticity of principal algebraic actions and Riesz product measures

Aihua Fan, Klaus Schmidt, Evgeny Verbitskiy

Abstract

For a continuous $\mathbb{N}^d$ or $\mathbb{Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb{Z}$ actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic $\mathbb{Z}^d$ ($d\ge 2$) actions of positive entropy under the condition of existence of summable homoclinic points.

Bohr chaoticity of principal algebraic actions and Riesz product measures

Abstract

For a continuous or action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic () actions of positive entropy under the condition of existence of summable homoclinic points.

Paper Structure

This paper contains 17 sections, 18 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Algebraic $\mathbb Z^d$-actions
  3. Mahler measure
  4. Main results
  5. $\mathbb Z^d$ actions of zero entropy
  6. Zero entropy $\mathbb{Z}^d$-actions are not Bohr chaotic
  7. Möbius disjointness and principal actions
  8. Riesz product measures on $X_f$
  9. Riesz product measures
  10. The dual group $\widehat{X}_f$
  11. Lacunary polynomials
  12. $(X_f, \alpha _f)$ is Bohr chaotic when $f$ is $m$-good
  13. Bohr chaoticity of $(X_f, \alpha _f)$: the case of $d=1$
  14. Bohr chaoticity of $(X_f, \alpha _f)$: the case of $d\ge 2$
  15. Homoclinic points of atoral polynomials in $R_d$ and the gap theorem
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose $f\in R_1\smallsetminus \{0\}$ with $\mathsf{m}(f)>0$. Then the principal algebraic $\mathbb Z$-action $(X_f,\alpha _f)$ is Bohr chaotic.

Theorems & Definitions (45)

  • Definition 1.1: FFS
  • Definition 1.2
  • Theorem 2.1
  • Definition 2.2: LSV
  • Theorem 2.3
  • Example 2.4: Toral automorphisms
  • Example 2.5: Constant polynomials
  • Example 2.6: The zero polynomial
  • Remark 2.7
  • Proposition 3.1
  • ...and 35 more