Non-spectrality Of Some Self-affine Measures On The Spatial Generalized Sierpinski Gasket

It has always been a hot topic whether the self-affine measure ?M,D is a spectral measure in the theory on spectral self-affine measure.On the spatial Sierpinski gasket,previous studies mainly focus on the spectrality and non-spectrality of?M,D with regard to diagonal matrices,upper triangular matrices and some special matrices.For more general expanding integer matrices,it is difficult to study the spectral property of self-affine measures.Inspired by the previous ideas,this paper discusses the non-spectral property of self-affine measures of some more general matrices.The main results are as follows:In the first part,for the expanding integer matrix M=[p1,p2,p3;p4,p5,p6;p7,p8,p9],where all the eigenvalues of M have modulus greater than 1,and pj?E Z(j=1,2,…,9),and the digit set D={0,e1,e2,e3},where e1,e2,e3 are the standard basis of unit column vectors in R3,drawing on the fact that the spectral property of self-affine measures are invariant under the similarity,we solve the non-spectrality of certain self-affine measures by choosing a suitable invertible matrix P to transform M into a diagonal matrix.In the second part,for the above digit set D,when the matrix M has three zero elements,we utilize a similar approach to the first part to solve the non-spectrality of some self-affine measures,which is also a supplement to previous achievements;when M=(p1,0,p1;0,p2,0;0,0,-p1],where p1 ?(2Z+1)\{±1} and p2?2Z\{0},by analysing the characteristics of the elements of the zero set Z(?M,D)and utilizing the principle of pigeon hole,we prove that ?M,D is a non-spectral measure and there exist at most 8 mutually orthogonal exponential functions in L2(?M,D).