Spectral Properties Of Self-affine Measure In Low Dimensional Spaces

If M ∈ Mn(R)is a real expanding matrix(that is,all the eigenvalues of the matrix M have moduli greater than 1),D C Zn is a finite digit set with cardinality |D|,let {Φd(x)}d∈D(Φd(x)= M-1(x + d),x∈Rn,d∈D)be an iterated function system(IFS).Then the IFS arises a natural self-affine measureμ:=μM,D satisfying μ＝1/|D|∑d∈DμOΦd-1,the measure μM,D is supported on the attractor(or invariant set)T(M,D)of IFS {Φd}d∈D.For a countable subset A C Rn,denote EΛ={e2πi<λ,x>:λ∈Λ},μ is a Borel probability measure with compact support on Rn,we call μ a spectral measure and Λ a spectrum of μ if EΛ is an orthogonal basis for L2(μ),and we say that(μ,Λ)is a spectral pair.The spectral problems of self-affine measure is a basic problem in harmonic analysis.In this thesis,we will mainly study the spectral problems of some self affine measures μM,D in low dimensional space.This problem originates from the conjecture of Fuglede in 1974,and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthogonal basis,but some do not.This thesis consists of three chapters.In Chapter 1,we firstly introduce the background and its significance of spectral and non-spectral problems of fractal self-affine measures,and we state the main results of this thesis.In Chapter 2,we mainly study the spectral problems of self-affine measureμb,D generated by product digit sets in one dimensional space,and get the following result:Let N,p,q∈N+,D={0,1,…,N-1}(?)Np{0,1,…,N-1},and p,q satisfy that p≥1,q≥2,and b=Nq,then μb,D is a spectral measure if and only if q(?)p.In Chapter 3,we mainly study the non-spectral problems of self-affine measure μM,D generated by expanding matrix M∈M2(Z)with three-elementdigit set(?).We prove that if det(M)(?)3Z,then μM,D has at most 81 mutually orthogonal exponentials in L2(μM,D),and the number "81" is the best.