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 μ₄ with exponential orthogo-nal basis E_∧={e^2πi<x,λ>:λ∈∧} on L²（μ）.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 μ_{pn},{Dn}（called Moran measure）generated by the following infinite convolution product μ_{pn},{Dn}=δ_p1-1D₁*δ{P₁P₂)-1D₂*…in the weak convergence,where*is the symbol of convolution and δe is the Dirac measure of this point e ∈ R.We show that if （?）,then μ_{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 μ_{pn},{Dn}.A real number p is called a spectral eigenvalue of μ if there exists a discrete set ∧ such that both A and pA are spectra forμ.We show that if p is a spectral eigenvalue of μ_{pn},{Dn},then p=p₁/p₂,where P₁,p₂ and 3 are pairwise coprime.