Non-spectrality Of A Class Of Planar Self-affine Measure And Signularity Of A Class Of Self-affine Measure In R~3

This thesis mainly investigates non-spectrality of a class of pla-nar self-affine measure with three-element digit set and lower bound estimate for the Fourier transform sequence of a class of self-affine measure in the generalized direction in R3. The main results are as follows:(1)By use of the Fourier transform of self-affine measure and Parseval equality,it is proved that under certain conditions,compatible pairs can be obtained by some spectral pairs.(2)By use of the residue class of 3 and the periodicity of the matrix operator M* when it acts on zero-point sets Z0 and Z0,it is proved that for the matrix and the digit set(a)if l(?)3Z+2,then there are no more than 2 mutually orthogonal exponential functions in L2((μM,D) which is determined by (M,D).(b)if l∈3Z+2,then there are at most 3 mutually orthogonal exponential functions in L2(μM,D) which is determined by (M, D). (3) By use of the properties of Pisot number and cosine function, it is proved that if a is a Pisot number,λ=α-1, the direction W={ξ(a1,a2,a3)t:ξ∈R, (a1,a2,a3)t∈Z3},then there exists a positive lower bound for the sequence{|μλ,D,W(αk)|}k=0∞and then the self-affine measureμλ,D is singular.The results of this thesis improve the related conclusions finished by D.E.Dutkay, P.E.T.Jorgensen,J.-L. Li, K.A.Kornelson and K.L.Shuman.