The Non-spectral Problem Of Sierpinski-type Self-affine Measure

It is a historic achievement in the cross-over study of harmonic analysis and fractal geometry is that one-fourth Cantor measure a(a singular and non-atomic fractal measure)admits an orthogonal exponential basis E?={e2?i<?,x>?? ?}for the space L2(?),which was found by Jorgensen.P.T.and Pederson.S[20]firstly.The measure ? with the above properties is called a spectral measure and ? is called a spectrum for ? This great discovery made the Fourier analysis for fractal sets to be a very hot topic in mathematics quickly.The problems of them had received a lot of attention in recent years.There are a series of excellent research results on the non-spectral prob-lem of the plane self-affine measure ?M,D(see definition(1.1)).But these results are all based on the integer expansion matrix M ? M2(Z),the skills are not easily migrated to the real expansion matrix.We consider the non-spectral problem of Sierpinski-type self-affifne measure It is generated by an real expansive matrix M=(?)and the three-elements digit where 0<|p|<1 and p ? S(p,q)={±(q/p)1/r:1 ?q<p,(p,q)=1,3 |p}.We know it is not a spectral measure,so how many it is the cardinality of E?={e2?i<?,x>:???} in L2(?M,D)?We consider this problem in two cases.On the one hand we show that there exist at most 3 mutually orthogonal exponential functions in L2(?p)and the number 3 is the best possible if 3 | q.On the other hand we show that the cardinality of EA is finite if 3 | q.But for any given positive integer N,we can find N mutually orthogonal exponential functions in L2(?).Hence the supreme of the numbers of orthogonal exponential functions in L2(?)is ?,i.e.n*(?M,D)=?(see definition(2.1.2)).This provides a research idea for studying the more general self-affine measure ?M',D',where M'=(?)? M2(R) and (?).