The Non-spectral Analysis Of Self-affine Measures On Spatial Sierpinski

Let M?Mn(Z)be an expanding matrix,D(?)Zn be a finite digit set of cardinality|D|.The self-affine measure ?M,D associated with M and D is the unique probability measure ? satisfying the self-affine identity with equal weight:?=(?),where ?d(x)=M-1(x+d)is an affine mapping.Its spectrality or non-spectrality has received much attention in recent years.This paper mainly considers finiteness of orthogonal exponentials in the Hilbert space L2(?M,D).The main contents of this paper are as follows:In the first part,let M=diag[p1,p2,p3]and D={0,e1,e2,e3},where p1,p2,p3 ?Z\{0,±1},e1,e2,e3 are the standard basis of unit column vectors in R3.The s-pactial Sierpinski gasket T(M,D)corresponding to the matrix M and the digit set D is a typical fractal on R3.Let ?M,D be the self-affine measure supported on T(M,D)with equal weight.In the case p1?2Z,p2,p3?2Z+1 and p2?p3,the finiteness of orthogonal exponentials in the Hilbert space L2(?M,D)has been proved,but the maximal cardinality of orthogonal exponentials is not known.With the same M and D,the constructed five-element and eight-element orthogonal exponentials in L2(?M,D)disprove the conjecture that the maximal cardinality of orthogonal expo-nentials is "4'.In the present paper,we continue to study the problem of maximal cardinality of orthogonal exponentials in L2(?M,D).In the case p2?±p3,we con-struct a class of maximal ten-element orthogonal exponentials in the corresponding Hilbert space L2(?M,D).This answers the question whether all eight-element or-thogonal exponentials in L2(?M,D)are maximal negatively.In the second part,we will mainly analys the spectrality of the affine-measures?M,D when M is a class of special expanding matrix in the space R3 and D={0,e1,e2,e1+e2},where e1,e2 are the unit column vectors in R3,we obtain that ?M,D is a non-spectral measure,and there are at most 4 mutually orthogonal exponential functions in L2(?M,D),the number "4" is the best.