Monatshefte für Mathematik

, Volume 169, Issue 3–4, pp 397–407 | Cite as

Spectral self-affine measures with prime determinant

Article

Abstract

The self-affine measure \(\mu _{M,D}\) relating to an expanding matrix \(M\in M_{n}(\mathbb Z )\) and a finite digit set \(D\subset \mathbb Z ^n\) is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of \(\mu _{M,D}\) in the case when \(|\det (M)|=p\) is a prime. The main result shows that under certain mild conditions, if there are two points \(s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}\) such that the exponential functions \(e_{s_{1}}(x), e_{s_{2}}(x)\) are orthogonal in \(L^{2}(\mu _{M,D})\), then the self-affine measure \(\mu _{M,D}\) is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.

Keywords

Iterated function system Self-affine measure Spectrality 

Mathematics Subject Classification (2010)

28A80 42C05 46C05 

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Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

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