Spectrality And Non-spectrality Of A Class Of Self-affine Measures With Two-element And Four-element Digit Sets

The self-affine measure μM,D is uniquely determined by the iterated function system {Φd（x）=M-1（x+d）}d∈D,where M ∈ Mn（Z）is an expanding matrix that all the eigenvalues have moduli greater than 1,and D（?）Zn is a finite digit set.The problem on the spectrality and non-spectrality of the self-affine measure μM,D（that is,the problem on the existence of orthogonal exponential basis in the Hilbert space L2（μM,D））has been received much attention and it’s a hot topic in recent years.Based on the previous results,this thesis investigates the spectrality and non-spectrality of self-affine measures with two-element digit set in Rn and a class of four-element digit set in the plane.The main conclusions are the following:（1）We study the spectrality of the self-affine measure μM,D corresponding to an expanding integer matrix M and the two-element digit set D={0,α}（?）Zn,α=（α1,α2,…,αn）t,（α12+α22+…+αn2≠0）in Rn with αiβj-αjβi≠0 by defining βγ=pγ1α1+pγ2α2+…+pγnαn=（γ=1,2,…,n）.Furthermore,when αis an eigenvector of matrix M with certain eigenvalue l,where l∈Z\{0,±1},we conclude:if l is odd,then there are at most 2 mutually orthogonal exponentials in L2（μM,D）;if l is even,then μM,D is a spectral measure.（2）For an expanding integer matrix M and the four-element digit set D={（0,0）t,（1,0）t,（2,0）t,（l,1）t｝,in the case when det（M）∈2Z,we first determine the finiteness and infiniteness of orthogonal exponentials in L2（μM,D）,and further obtain some results about the spectrality and non-spectrality of μM,D by using the residue system of modulo 4 for the matrix M,and by analyzing the characteristic of the zero set of symbol function mD（x）in[0,1)2.We consider the problem on the spectrality and non-spectrality of the self-affine measure with two-element digit set,which generalizes several conclusions of Li and Wen[17]from two dimensions to higher dimensions.This research perfects the corresponding results in high dimensions.Besides,Li[22]and Liu[28]research the spectral property and non-spectral property of the self-affine measure corresponding to an expanding integer matrix M with det（M）∈ 2Z and the four-element digit set D={（0,0）t,（1,0）t,（2,0）t,（l,1）t} with l=0.In the present thesis,we improve the results of Li[22]and Liu[28],and extend related results to the case where l is any integer.