Spectral Researchs Of Self-affine Measures

In fractal geometry combined with harmonic analysis,a brilliant achie-vement is the 1/4 Cantor measure ? with exponential orthogonal basis EA={e2?i<x,?>,? ? ?} on L2(?)which was found firstly by Jorgensen and Peder-son[11].We call spectral measure which satisfies the above properties,and the spectrum is A.This amazing discovery quickly made Fourier analysis on frac-tals to be a hot topic of mathematics.People study around common singular measures to determine their spectral properties.This paper is to study the spectrality of fractal measures,which consists of two parts.The first part is to study Moran measure,which determines that such a measure is a spectral measure and found the specific form of the spectrum A.We consider a one-dimensional triple integer number set Dn ={0,an,bn},n ? 1,and positive integer sequences pn satisfying the following conditions:supn?1{|an|/pn.|bn|pn}<?.For the above sequence,it is already known that there exists a unique Borel probability measure ?{pn},{Dn}(called Moran measure)generated by the following infinite convolution product?{pn},{Dn}=?p1-1D1*?(p1p2)-1D2*...in the weak convergence,where ?e is the Dirac measure of this point e ? R and*is the symbol of convolution.We give a sufficient condition in which?{pn},{Dn},is a spectral measure,and the specific form of the spectrum.The second part of the paper is to study the spectral property of self-affine measure ?,?,D(the definition is in(1.1.1)).We consider a digit set D ={0,1,2,3,4,5} and the contraction ratio ?=(1/5)1/r,r>2 and r ? Z.It is known that such a self-affine measure ?(1/5)1/r,D is not a spectral measure.A na-tural question is that what is the number of the largest exponential orthogonal systems EA?In the case of r = 2,3,we have proved that the maximum num-ber#D of mutually orthogonal exponential systems in L2(?(1/5)1/r,D)is 6,and the number 6 is not modifiable.It provides ideas for studying the spectrality of general self-affine measure ?±(q/p)1/r,D2,D2={0,1,...,N-1},?2=±(q/p)1/r.