自仿测度的非谱准则

Non-Spectral Criterions for Self-Affine Measures

设μ_(M,D)是由仿射迭代函数系{φ_d(x)=M^(-1)(x+d)}_(d∈D)唯一确定的自仿测度,它的谱性或非谱性与Hilbert空间L^2(μ_(M,D))中正交指数基(也称为Fourier基)的存在性有着直接的关系.近年来自仿测度μ_(M,D)的谱性或非谱性问题的研究受到人们普遍的关注.本文给出了判定自仿测度μ_(M,D)非谱性的几个充分条件,所得结果改进推广Dutkay,Jorgensen等人的非谱准则.

Let μM,D be the self-atone measure uniquely determined by the iterated function system （IFS）{Фd（χ） = M-1 （χ ＋ d）}d∈D with equal weight, where M ∈ Mn（R） is an expanding matrix and D C Rn is a finite digit set. The spectrality or non- spectrality of μM,D is directly connected with the existence of orthogonal exponential basis （Fourier basis） in the Hilbert space L2（μM,D）, and has been received much atten- tion in recent years. In this paper, we provide several sufficient conditions for self-affine measures to be non-spectral. The results here extend the corresponding non-spectral criterions of Dutkay, Jorgensen and others in a simple manner.