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

Let ? 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(?):={e2?i<?,x>:???} forms an orthonormal basis for L2(?).In this thesis,we mainly study the spectrality of self-affine measure ?M,D generated by an integer expanding matrix M ? M2(Z)and integer digit set D={(0,0)t,(?1,?2)t,(?1,?2)t},where det(M)? 3Z and?1?2-?2?1?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 ?M,D for case 1:2?1?1,2?2-?2?3Z and case 2:2?1-?1?3Z or 2?2-?2?3Z respectively.We obtain that ?M,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 2?1-?1,2?2-?2?3Z,?M,d is spectral measures if and only if L2(?M,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.