Self affine Delone sets and deviation phenomena

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.

42 pages

References in corpus (1)

• Pure Point Dynamical and Diffraction Spectra

Cited by in corpus (6)

• Random Substitution Tilings and Deviation Phenomena
• Rotation numbers and rotation classes on one-dimensional tiling spaces
• Canonical diffusions on the pattern spaces of aperiodic Delone sets
• Tilings and traces
• Traces of random operators associated with self-affine Delone sets and Shubin's formula
• Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals