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.
Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them.
LI JianLin College of Mathematics and Information Science,Shaanxi Normal University,Xi’an 710062,China
