具有2个数字集的自仿测度μM,D正交指数函数的个数

The cardinality of μM,D orthogonal exponentials of self-affine measures with two-elements digit set

对于由M=pIN(|p|>1,p∈Z),D={0,l1e1+l2e2+…+lNeN}ZN(l21+22+…+l2N≠0,lj∈Z,j=1,2,…,N)决定的自仿测度μM,D,支撑在吸引子T(M,D)上.证明当p为奇数时,L2(μM,D)空间中的正交指数函数系最多有2个元素,而且2是最好的估计;当p为偶数时,L2(μM,D)空间中存在含有无限个元素的正交指数函数系.

The self-affine measure μM.D corresponding toM--PIN M=pIN（｜p｜〉1,p∈Z）,D={0,l1e1＋l2e2＋…＋lNeN}ZN（l21＋22＋…＋l2N≠0,lj∈Z,j=1,2,…,N） is supported on the attractor T（M,D）.It was showed that there exist at most 2 mutually orthogonal exponential functions in L2 （μM, D） when p is odd;there exist infinite mutually orthogonal exponential functions in L2 （μM,D） when p is even.