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Pure point diffraction and entropy beyond the Euclidean space

Till Hauser

Abstract

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of sigma-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set can not be computed along a limit and even be infinite in this setting.

Pure point diffraction and entropy beyond the Euclidean space

Abstract

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of sigma-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set can not be computed along a limit and even be infinite in this setting.

Paper Structure

This paper contains 22 sections, 15 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally compact Abelian groups
  4. Haar measure
  5. Van Hove sequences
  6. Delone sets
  7. Finite local complexity and uniform patch frequencies
  8. Pure point diffraction
  9. Cut and project schemes and regular model sets
  10. Actions of LCA groups
  11. Uniformity
  12. Local rubber topology
  13. Delone actions
  14. Topological entropy of actions
  15. Measure theoretical aspects of continuous dynamics
  16. ...and 7 more sections

Key Result

Proposition 1.1

If $G$ is a LCA$_\sigma$ group, then the topological entropy of a Delone subset $\omega$ is given by where $(A_n)_{n\in \mathbb{N}}$ is a van Hove sequence in $G$ and where the supremum is taken over all open neighbourhoods of $0$.

Theorems & Definitions (29)

  • Proposition 1.1
  • Lemma 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Example 2.5
  • Proposition 2.6
  • Theorem 2.7: Halmos- von Neumann
  • Proposition 2.8
  • Lemma 3.1
  • ...and 19 more