The Spectrality Of Self-affine Measures On Sierpinski Gasket

For an affine iterated function system {?d(x)= M-1(x+d)}d?D,where M is an expanding integer matrix and D is a finite integer digit set,there exists a unique probability measure?:??M,D satisfying self-affine identical equation?=1/|D|?d?D?o?-1d.The self-affine measure is an unique measure.The theory of self-affine measure mainly studies the spectrality and non-spectrality of the self-affine measures ?,M,D The question for the planar spectrality of self-affine measures is clear,where D is a standard digit set,D = ?0,e1,e2?,e1 =(1,0)t,e2 =(0,1)t.But if self-affine measure is a spectral measure,the question is whether we can find all spectrum.Due to the rise of dimension,it is difficult to deal with the relations among the zero-sets in R3,so only partial results about the spectrality of self-affine measures.In this paper we divided into two parts to discuss the spectrality of self-affine measures,the main results are as follows:In the first part,we research the self-affine measure ?M,D corresponding to a symmetric matrix M and the digit set D = {0,e1,e2,e3 in R3 is supported on the generalized spatial Sierpinski gasket,where e1,e2,e3 are the standard basis of unit column in R3.By doing similarity transformation into a manageable form and the spectrality of self-affine measure is invariant under Z-similarity,we show that the?M,D is a spectral measure.In the second part,we deal with the problem is how to constitute the spectrum of a spectral self-affine measure.First we use two methods shows that a theorem gives an error spectrum.The second we use different methods to give this case many of the spectrum,and the spectrum is different,so complete the self-affine measure results.