R ~ N In Particular Spectrum And Spectral Properties Of The Non-self-imitation Measure

This paper mainly reasearch about the spectrality and non-spectrality of a class of self-affine measure μM,D which is decided by a class of expanding integer diagonal matrix or a class of expanding integer triangular matrix and digital set with two elements, And the spectrality and non-spectrality of a class of self-affine measure μM,D which is deretated by a class of expanding integer diagonal matrix or a class of expanding integer triangular matrix and digital set with decomposable form.The main result of the paper as follows(1)By using characteristics of the zero set Z(μM,D),the knowledge of compatible pair and Strichartz’ theorem proving on the spectrality proved the spectrality and non-spectrality of a class of self-affine measure μM,D is decided by a class of expanding integer diagonal matrix M and digital set D with two-elements.(2) By using the knowledge of compatible pair, Strichartz’ theorem proving on the spectrality and the situation of expanding integer diagonal matrix proved the spectrality and non-spectrality of a class of self-affine measure μM,D which is decided by a class of expanding integer triangular matrix M and digital setD with two-elements.(3)Proved the spectrality and non-spectrality of a class of self-affine measure μM,D D which is decided by a class of expanding integer diagonal matrix M and digital set D with decomposable form with the spectrality and non-spectrality of self-affine measure which digital set with two-elements and the knowledge of direct sum.(4)With the spectrality of self-affine measure which digital set with two-elements and the knowledge of direct sum proved that the self-affine measure which is decided by a class of expanding integer triangular matrix M and digital set D with decomposable form is a spectral measure with the spectrum in the decomposable form.Jorgensen, Pedersen, J.-L. Li, Wen ZhiYing and others have deeply studied the spectrality and non-spectrality of self-affine measure which digital set with two-elements in the case of one-dimensional and two-dimensional, J.-L. Li and others further studied the spectrality and non-spectrality of self-affine measure which digital set with decomposable form in the case of two-dimensional by using the knowledge of direct sum. This paper studied the spectrality and non-spectrality of self-affine measure which digital set with two elements and decomposable form, it is an effective promotion of the results of the previous. The results would have a good effect on the related study in future.