The Study Of Orthogonal Exponential Sets Of Two Kinds Of Fractal Measures

In this paper,we mainly study the spectral problem of a class of self-affine measures ?_M,D and a class of Moran measures ?_{Rk},{Dk}.This paper is divided into three chapters,which are specifically arranged as follows:In Chapter 1,we first introduce some basic knowledge needed in this paper,then introduce the research background and status of spectral theory of fractal measures,and finally list the main conclusions of this paper.In Chapter 2,we mainly study the non-spectral problem of the self-affine measure ?_M,D in Rⁿ. ?_M,D is generated by the real diagonal matrix M and the digit set D (?) Rⁿ satisfying (?)_dⁿ(?)(?)_mⁿ,((?)_Dⁿ-(?)_Dⁿ)\{0} (?)(?)_Dⁿ(mod Zⁿ),where (?)_Dⁿ:={x?[0,1)ⁿ:m_D(x)=0} and Emn:=1/m{(l₁1,l₂,…,l_n)t:0:?l₁,l₂,…,l_n?m-1}\{0}.If L²(?_M,D)only has finite orthogonal exponential functions,we give an estimate of the exact maximal cardinality.In Chapter 3,we mainly consider the Moran measure ?_{Rk},{Dk} generated by an expanding matrix sequence {R_k}_k=1^? (?) M_n(Z) and a sequence of finite integer digit sets {D_k=B_kD}_k=1^?.If (?)_Dⁿ=(?)_pⁿ,B_k?M_n(Z) and gcd(det(B_k),p)=1,then L²(?_{Rk},{Dk})contains infinite families of orthogonal exponential functions if and only if there exist subsequences {k_i}_i=1^?(?)N such that gcd(det(R_ki),p)>1.