The Finiteness Of Orthogonal Exponentials On The Sierpinski Gasket

Let M ? Mn(Z)be an expanding integer matrix and D(?)Zn a cardinality of |D| finite digit set.The self-affine measure ?M,D determined by the iterated function system(IFS){?d(x)=M-1(x + d)}d?D is a unique probability measure satisfying self-affine identical equation ?=1/|D| ?d?d ?d-1,and its spectrality and non-spectrality have become the important research contents in the self-affine measure theory.In this part,we mainly study the first class problem of the non-spectrality of the self-affine measure on the Sierpinski gasket.That is,there are at most finite orthogonal exponential functions in Hilbert space L2(?M,D).And for the finite problem of the orthogonal exponential system needs to be solved on the Sierpinski gasket in the plane and space,The partial results are given in the two part.The main results are the following:In the first part,for the general form of the space Sierpinski gasketT(M,D),where M=diag[p1,p2,p3](p1,p2,p3 ?Z\{0,±1}),D={0{e1,ee2,e3}},and e1,e2 and e3 are the standard basis of unit column vectors in R3,the finiteness or infiniteness of self-affine measure ?M,D has been solved completely.But in the finite case,when p1 is even,p2 and p3 are odd and p2 ? p3,the finiteness of the orthogonal exponentials on space L2(?M,D)is only given guess:its best upper bound is "4".In this part,we construct a series of five elements orthogonal exponential function system in the Hilbert space L2(?M,D).It is proved that the guess about the above best upper bound is wrong,which provides a basis for further study of its finite problem.In the second part,for the plane Sierpinski gasket,it is known that the spec-trality and non-spectrality of the self-affine measure ?M,D has been solved.And in the non-spectral problem,the optimal number of the orthogonal exponentials in the Hilbert space L2(?M,D)has also been determined.But the general form need-s to be researched further.In this part,some results are obtained based on the characteristics of the zero set of Fourier transform.