Non-spectrality Of Two Classes Of Self Affine Measures

Let ?M,D be the self-affine measure uniquely determined by an expanding matrix M ? Mn(Z)and a finite digit set D(?)Zn through the affine iterated function system(IFS){?dd(x)= M-1(x+d)}d ?D + In this paper,we mainly study the problem of class I in the non-spectrality of self-affine measure:there are at most a finite number of orthogonal exponential functions in Hilbert space L2(?M,D),and the optimal upper bound of the number of orthogonal exponential functions in L2(?M,D)is estimated.In addition,we have made some research on the non-spectrality of special planar self-affine measure with four-elements digit set.The main contents of this paper are as follows:In the first part,the non-spectrality of ?M,D is directly connected with the finite-ness or infiniteness of orthogonal exponentials in the Hilbert space L2(?M,D).In this part,we provide a better estimate on the cardinality of ?M,D-orthogonal exponentials by characterizing the zero set Z(mD)of the symbol function mD(x)and its middle points.The results here extend the corresponding results of Dutkay?Jorgensen and others.In the second part,the spectrality and non-spectrality of planar self-affine mea-sure with three-elements digit set has been completely resolved,but the spectrality and non-spectrality of planar self-affine measure with four-elements digit set has not completely solved.In this part,we obtain some results on the non-spectrality of special planar self-affine measure with four-elements digit set.