Dynamical versus diffraction spectrum for structures with finite local complexity

Abstract

It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

27 pages; some minor revisions and improvements

References in corpus (6)

• Pure Point Dynamical and Diffraction Spectra
• Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations
• Dynamical properties of almost repetitive Delone sets
• The singular continuous diffraction measure of the Thue-Morse chain
• Examples of substitution systems and their factors
• Close-packed dimers on the line: diffraction versus dynamical spectrum

Cited by in corpus (13)

• Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction
• Ergodic properties of visible lattice points
• Aperiodic crystals and beyond
• Spectral analysis of a family of binary inflation rules
• Examples of substitution systems and their factors
• Spectral Theory of Substitutions in ${\mathbb{Z}}^d$
• Dynamical properties of $k$-free lattice points
• Lyapunov exponents for binary substitutions of constant length
• Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem
• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions
• Diffraction Theory of Point Processes: Systems with Clumping and Repulsion
• Recent progress in mathematical diffraction
• Rauzy fractals of random substitutions