The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.
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.
