The Study Of Spectral Eigenvalues Of Two Kinds Of Moran Measures

Fractal geometry is a popular research discipline,which has extensive crossover and integration with many other disciplines.In recent years,Fourier analysis on Fractal has become a hot research topic.Fractal spectral measure is a basic problem on the study of Fourier analysis on Fractal.Let ?be a Borel probability measures on Rn.A real number d is called a spectral eigenvalue of? if there exists a discrete set ? such that both A and dA are spectra for ?.In this paper,we will mainly study the spectral eigenvalues of two kinds of Moran measures.The main work of this paper is spread in the following three chapters:In Chapter 1,we introduce the research background and current situation of fractal spectral measure.Simultaneously,we will recall some basic concepts,and give several theorems and lemmas which will be needed.At last we give our main results of this paper.In Chapter 2,we mainly study the spectral eigenvalue of a class of Moranmeasures ?_?,{Dn} generated by positive integer ?=N^s+1 and direct sum consec-utive digit set D_n={0,1,…,N-1}???N^t_nl_n{0,1,…,N-1},where s,N?2 are positive integers and {t_n}_n=1^?,{l_n}_n=1^? are sequences of integers with bound-ed with gcd?l_n,N?=1.If s????{t_n},then we obtain a sufficient and necessarycondition for the real number d to be spectral eigenvalues of ?_?,{Dn}.In Chapter 3,we mainly study the spectral eigenvalue of planar Moran mea-sures ?_M,{Dn} generated by an expanding integer matrix M=???and the four-element integer digit set D_n={?0,0?^t??a_n,0?^t,?0,b_n?^t,?±a_n,±b_n?^t}:=D_n⁺ or D_n^-where a_n?b_n are bounded and p,q ? Z ?[2,+?).For arbitraryn?1 satisfying 2|p/gcd?a_n,p?,2|q/gcd?b_n,q?,we get a sufficient and necessary condi-tion for the real number d to be spectral eigenvalues of ?₍M,{D_n^-}_....