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Communications in Mathematical Physics

, Volume 357, Issue 3, pp 1071–1112 | Cite as

Self Affine Delone Sets and Deviation Phenomena

  • Scott Schmieding
  • Rodrigo Treviño
Article

Abstract

We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA
  2. 2.University of MarylandCollege ParkUSA

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