Spectrality Of Some Special Self-affine Measures

Let μM,D be a self-affine measure generated by the affine iterated function system {φd(x)=M-1(x+d)}d∈D,where M is a n x n real expanding matrix and D?Rn is a finite digit set.The research on the spectrality of a self-affine measureμM,D is to investigate whether there exists an orthogonal basis consisting of exponential functions in Hilbert space L2(μM,D).Nowadays the study of the spectrality or nonspectrality of μM,D has attracted much attention and achieved considerable results,but there are still many problems to be solved by further research.This thesis mainly focus on the conditions under which the self-affine measure μM,D corresponding to M and D is a spectral measure.The main work is as follows:1.We study the spectrality of self-affine measures μM,D in the case when |det(M)|=|D|=p is a prime.Firstly,we obtain two more precise and easily verifiable sufficient conditions for μM,D to be a spectral measure with lattice spectrum.Secondly,by the Hermite normal forms of M2 and M*2,we provide the form of elements in the digit set D and the zero set Z(μM,D)that cannot satisfy these two sufficient conditions respectively.Finally,we discuss the application of these two sufficient conditions in integral self-affine tiles,which provides new insights into a conjecture of Lagarias and Wang.2.We focus on the spectrality of self-affine measures μM,D and digit set D related to μM,D in the case when |det(M)|=pα(α=1,2,…)is a prime power and |D|=p is a prime.We establish the relation between the orthogonal system and the structure of vanishing sums of roots of unity by the pα-th cyclotomic polynomial.With the help of the property of a residue system in number theory，we provide a new method to construct countably many sets S?Zn such that(M-1D,S)is a compatible pair,which implies that ΜM,D is a spectral measure and the related digit set D is a spectral set.When |det(M)|=2p(p≥3 is a prime)and |D|=p is a prime,we also prove that the digit set D corresponding to μM,D is a spectral set by the 2p-th cyclotomic polynomial.3.We consider the spectrality of two classes of self-affine measures.For a 3 x 3 integer expanding matrix M,when D?Z3 is a finite digit set with |D|=4,we obtain many conditions for(M-1D,S)to be a compatible pair,where S?Z3.Therefore,μM,D is a spectral measure.This is based on the characteristics of nonzero middle points in zero set Θ0.For a n x n integer expanding matrix M,when D?Zn is a finite digit set with |D|=p,we also prove that the corresponding self-affine measureμM,D is a spectral measure by the p-th cyclotomic polynomial.4.We investigate the spectrality of a class of self-affine measures with collinear digit sets.For the general form of the characteristic polynomial of integer expanding matrix M,we analyze the relation between its roots and coefficients,and then find out a spectrum for μM,D by constructing a appropriate similarity transformation based on the simple operation program in Matlab.Moreover,with the help of the characteristic polynomial coefficients of M,under a mild condition,we also obtain a necessary and sufficient condition that the self-affine measures with three-element collinear digit set are spectral measures.Finally,we solve a problem left by Liu et al.about the case of three-element collinear digit set on the plane.The results here extend the corresponding known results related to the spectrality of self-affine measures with collinear digit set.The above results only involve the research on the spectrality of some special selfaffine measures.We hope to shed new light on the study of partial problems about the spectral theory of self-affine measures.