Dynamical properties of almost repetitive Delone sets

Abstract

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.

arXiv comment28 pages, 1 figure, introduction expanded, some proofs adjusted, accepted for publication in Discrete and Continuous Dynamical Systems A

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Cited by in corpus (8)

• Dynamical versus diffraction spectrum for structures with finite local complexity
• Ergodicity and dynamical localization for Delone-Anderson operators
• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions
• A dichotomy for bounded displacement equivalence of Delone sets
• Multiscale Substitution Tilings
• Linear repetitivity beyond abelian groups
• Discrepancy and rectifiability of almost linearly repetitive Delone sets
• Substitution tilings with dense tile orientations and n-fold rotational symmetry