Spectral Property Of Certain Moran Measures With Three-element Digit Sets

Fractal geometry has penetrated into all branches of mathematics,especi-ally,great achievements have been made on the cross-research of fractal geo-metry and harmonic analysis.For instance:Jorgensen and Pederson[44]first discovered a singular,non-atomic fractal measure ?₄ with exponential orthogo-nal basis E_?={e^2?i<x,?>:???} on L²???.This amazing discovery quickly made the Fourier analysis on fractal sets to be a hot topic in mathematics.We call spectral measure which satisfies the above properties?and the spectrum is?.This thesis mainly studies the Moran measure,which consists of two parts.The first part is to study the spectral property of certain Moran measures with three-element digit sets.We consider a one-dimensional triple integer num-ber set D_n={0,a_n,b_n}={0,1,2}?mod 3?,and positive integer sequences pn satisfying the following conditions sup_n?1{|a_n|/p_n,|b_n|/p_n}<?.For the above se-quence,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*is the symbol of convolution and ?e is the Dirac measure of this point e ? R.We show that if ???,then ?_{pn},{Dn} is aspectral measure,and the specific form of the spectrum is given.The second part of this thesis is to study the spectral eigenvalue problem of this kind of measure ?_{pn},{Dn}.A real number p is called a spectral eigenvalue of ? if there exists a discrete set ? such that both A and pA are spectra for?.We show that if p is a spectral eigenvalue of ?_{pn},{Dn},then p=p₁/p₂,where P₁,p₂ and 3 are pairwise coprime.