The Cardinality Of Orthogonal Exponentials With Certain Planar Self-affine Measure

Let M be an expanding integer matrix,that is,all the eigenvalues of the integer matrix M have moduli>1,M*denotes the transposed conjugate of M.On the basis of previous studies,this thesis finds that M*j(j ? Z,j>1)has a certain rule when the elements in M*are classified according to the residue classes of module 3 or module 4,thus the zero set of the Fourier transform of the self-affine measure determined by the expanding matrix M and a class of planar digit sets D can be simplified.Finally,the non-spectrality of the self-affine measure ?M,D be obtained.We have the following results in this thesis:In the first part,we study the cardinality of orthogonal exponentials on the space L2(?M,D)determined by the expanding matrix M and a class of digit sets D in the plane when det(M)(?)3Z and D=D1(?)PD1 where After classifying the elements in M*according to the residue classes of module 3,we find that there are nine mutually orthogonal exponential functions on the space L2(?M,D)by analyzing the characterization of M*j(j ? Z,j>1)and the zero set of the Fourier transform of the self-affine measure ?M,D.In the second part,we discuss the non-spectrality of the self-affine measure?M,D determined by the following expanding matrix and the digit set After classifying the elements in M*according to the residue classes of module 4,we find that the cardinality of orthogonal exponentials on the space L2(?M,D)determined by the pair(M,D)is four or sixteen when det(M)? 2Z+1,that is,?M,D is non-spectral measure.