Non-spectral Conditions For Two Classes Of Self-affine Measures

Let M ? Mn(Z)be an expanding matrix and D(?)Zn be a finite digit set.The iterated function system {?d(x)=M-1(x+d)}d?D,x ?Rn defined by M and D can uniquely determine a self-affine measure ?M,D In this paper,we make use of different ways to study non-spectral conditions for two classes of self-affine measures by analyzing the characterization of the zero set Z(?M,D)which is the Fourier transform of the self-affine measure.The main results are as follows:In the first part,the iterated function system with two-element digit set is the simplest case and the most important case.The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.However,the higher dimensional analogue,especially the two-dimensional case has not been solved completely.Also,there is a conjecture to illustrate that in the plane,the remaining cases correspond to non-spectrality of self-affine measures.Motivated by this problem,we provide in this paper some non-spectral conditions for the planar self-affine measures with two-element digit set.Under one of the conditions,we determine the maximal cardinality of orthogonal exponentials.An application of this result and the validity of the conditions are also presented.In the second part,we mainly study the self-affine measure ?M,D where the zero set Z (?) Qn in the unit interval[0,1)n of the function mD(x)is finite.In order to explore the conditions for the corresponding Hilbert space L2(?M,D)including finite orthogonal exponentials,we provide a better estimate on the cardinality of orthogonal exponentials and determine the maximal cardinality of orthogonal exponentials under special conditions.The results here based on previous studies extend the planar results of Liu,Dong and Li to higher dimension.