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arXiv · 2001.10725

Linear repetitivity beyond abelian groups

Siegfried Beckus·Tobias Hartnick·Felix Pogorzelski
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Abstract

We show that linearly repetitive weighted Delone sets in groups of polynomial growth have a uniquely ergodic hull. This result applies in particular to the linearly repetitive weighted Delone sets in homogeneous Lie groups constructed in the companion paper arXiv:2109.15210 using symbolic substitution methods. More generally, using the quasi-tiling method of Ornstein-Weiss, we establish unique ergodicity of hulls of weighted Delone sets in amenable unimodular lcsc groups under a new repetitivity condition which we call tempered repetitivity. For this purpose, we establish a general sub-additive convergence theorem, which also has applications concerning the existence of Banach densities and uniform approximation of the spectral distribution function of finite hopping range operators on Cayley graphs.

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Siegfried Beckus, Tobias Hartnick, Felix Pogorzelski. 2020-01-29. Linear repetitivity beyond abelian groups. https://doi.org/10.1017/etds.2024.120

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