The Study Of Spectrality Of The Planar Self-affine Measures With Arbitrary Three-element Digit Sets

Let μ be a Borel probability measure with compact support on Rn.We call it a spectral measure if there exists a countable subset ∧? Rn such that the family of exponential functions E(∧):={e2πi<λ,x>:λ∈∧} forms an orthonormal basis for L2(μ).In this thesis,we mainly study the spectrality of self-affine measure μM,D generated by an integer expanding matrix M ∈ M2(Z)and integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t},where det(M)∈ 3Z andα1β2-α2β1∈3Z\{0}.This thesis is divided into three chapters:In chapter 1,we first introduce the research background and the research results at internal and abroad,and then introduce the main problems and list the main conclusions of this thesis.In chapter 2 and 3,we study the spectrality of μM,D for case 1:2α1β1,2α2-β2∈3Z and case 2:2α1-β1?3Z or 2α2-β2?3Z respectively.We obtain that μM,D is spectral measures if and only if there exists Q ∈ M2(R)such that(M,D)is admissible,where M=QMQ-1,D=QD.In particular,when 2α1-β1,2α2-β2∈3Z,μM,d is spectral measures if and only if L2(μM,D)contains an infinite orthogonal set of exponential functions.At the end of this thesis,we give two examples and propose a problem related to our study.