Self affine Delone sets and deviation phenomena

TL;DR

This paper investigates how the norms of ergodic integrals grow in spaces derived from self-affine Delone sets, linking growth rates to cohomological actions and exploring implications for diffraction and specific examples.

Contribution

It introduces a framework connecting growth of ergodic integrals to cohomological dynamics in self-affine Delone sets and analyzes diffraction properties for substitution and cut-and-project sets.

Findings

Growth rates are governed by the induced cohomological action.

Diffraction properties are influenced by the self-affine structure.

Explicit examples demonstrate the theoretical results.

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.