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 EΛ={e2πi<λ,x>λ∈ Λ}for the space L2(μ),which was found by Jorgensen.P.T.and Pederson.S[20]firstly.The measure μ with the above properties is called a spectral measure and Λ is called a spectrum for μ 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 μM,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 EΛ={e2πi<λ,x>:λ∈Λ} in L2(μM,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(μp)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(μ).Hence the supreme of the numbers of orthogonal exponential functions in L2(μ)is ∞,i.e.n*(μM,D)=∞(see definition(2.1.2)).This provides a research idea for studying the more general self-affine measure μM’,D’,where M’=(?)∈ M2(R) and (?).