Spectrality And Non-spectrality Of A Class Of Self-affine Measures With Four-element And Three-element Digit Sets

The self-affine measure μM,D associated with an expanding matrix M ∈Mn(Z) and a finite digit set D C Zn is uniquely determined by the self-affine identity with equal weight. In 1998 Jorgensen and Pedersen found an example of the self-affine measure is spectral for the first time, to promote the concept of spectral set. Since then many scholars have focused on the issues of spectrality and non-spectrality of self-affine measures. On the basis of their researches, the main results of the paper are as follows:In the first part, we construct a class of self-affine measures μM,D with four-element digit sets in the higher dimensions (n> 3) such that the Hilbert space L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane. Then, the spectrality of a class of self-affine measures μM,D with two-element digit sets are discussed.In the second part, the spectrality and non-spectrality of self-affine measures are invariant under the similarity. Firstly, by use of the invariance property and the Strichartz’s theorem, we discuss the spectrality of self-affine measures μM,D corre-sponding to a lower-triangle expanding integer matrix M∈E M3(Z) and a collinear digit set with three elements D(?)Z3. Secondly, we consider the spectrality and non-spectrality of a class of self-affine measures μM,D corresponding to an expanding integer matrix M ∈M3(Z) and a three-element digit set D(?)Z3 with parameters. Finally, we show that if p1(?)3Z, M∈M3(Z) is an upper-triangle expanding matrix and D={0,e1,e2+e3}, where e1=(1,0,0)t, e2=(0,1,0)t,e3=(0,0,1)t, then the corresponding self-affine measures μM,D is not spectral measure, and there exists at most 3 mutually orthogonal exponentials in L2(μM,D).