A Condition For The Finite Exponential Orthogonal System And Non-spectrality Of Self-affine Measures

Let μM,D be the self-affine measure uniquely determined by the iterated function system(IFS){φd(x)= M-1(x+d)}d∈D, where M ∈ Mn(Z) is an expangding matrix and D (?) Zn is a finite digit set. When the zero set Z(mD) in [0,1)" is finite, the digit set D is special. Therefore, we can make use of different ways to study spectrality and non-specrality of the self-affine measures μM,D·We have the following research results for this type of self-affine measure in this paper:In the first part, we mainly provide a sufficient condition for the finite μM,D-orthogonal exponentials by applying the elementary transformations of a matrix, which is based on the sufficient condition of Hilbert space L2(μM,D) including finite orthogonal exponentials proved by Dutkay and Jorgensen. This sufficient condition depends only upon the determinant of the matrix M, and is easy to use in the research of non-spectrality of μM,D·In the second part, we first give a simple and convenient way of finding com-patible pair based upon the research of predecessor. The advantage of the way is simplifying solving multiple equation sets. In addition, this way still apply to the case where zero set in [0,1)n of function mD(x) is finite. Secondly, we supplement the theorem in the first part so as to it can be used more conveniently. Finally, we summarize some ways of studying spectrality and non-specrality of the self-affine measures using an example.