摘要
For an expanding integer matrix M ∈ Mn(Z) and two finite digit sets D,S Zn with 0 ∈ D ∩ S,we shall investigate and study the possible conditions on the spectral pair(μM,D,Λ(M,S)) associated with the iterated function systems {φd(x) = M-1(x + d) }d∈D and {ψs(x) = M-x + s}s∈S in the case when |D| = |S| = | det(M) |. Under the condition that(M-1D,S) is a compatible pair,we obtain a series of necessary and sucient conditions for(μM,D,Λ(M,S)) to be a spectral pair. These conditions include how to characterize the invariant sets Λ(M,S) and T(M,D) such that Λ(M,S) = Zn and μL(T(M,D)) = 1 which play an important role in the number system research and in the construction of Haar type orthogonal wavelet basis respectively.
For an expanding integer matrix M ∈ Mn（Z） and two finite digit sets D, S C Z^n with O∈ D ∩ S, we shall investigate and study the possible conditions on the spectral pair （μM,D,A（M, S）） associated with the iterated function systems {Фd（X） = M^-1（x ＋ d）}d∈D and {φs（x） = M^＊x ＋ s}s∈S in the case when ｜D｜= ｜S｜ = ｜ det（M）｜. Under the condition that （M^-1D, S） is a compatible pair, we obtain a series of necessary and sufficient conditions for （μM,D, A（M, S）） to be a spectral pair. These conditions include how to characterize the invariant sets A（M, S） and T（M, D） such that A（M, S） = Zn and μL（T（M, D）） = 1 which play an important role in the number system research and in the construction of Haar-type orthogonal wavelet basis respectively.
基金
supported by the Key Project of Chinese Ministry of Education (Grant No. 108117)
National Natural Science Foundation of China (Grant No. 10871123, 11171201)
