Spectrality And Non-spectrality Of Certain Self-affine Measures On R~3

The research on self-affme measure μM,D is one of the important subject in fractals. The spectrality or non-spectrality of μM,D has received much attention in recent years. For the typical fractals such as the Sierpinski gasket in the plane R² and in the space R³, the corresponding spectrality and non-spectrality have been established. The generalized results for all fractals are difficult to obtain. and it need much work to do. It is known that the Sierpinski gasket in the space R³ is different from the case in the plane R². With the increase of dimensions, the difficulty also gradually becomes larger. In this thesis, we mainly deal with the spectrality and non-spectrality of self-affine measures in the case when M is a special expanding matrix. The content of this thesis has two parts. The main results are the following.In the first part, based on the fact that spectral pair and compatible pair are invariant under the similarity, we present a method of dealing with the spectral-ity and non-spectrality of self-affine measures. The method shows how to choose a reversible matrix P reasonably, so that M and the zero set Z{μM,D ) of the Fourier transform μM,D are simplified simultaneously. The simplified form enables us to deal with the spectrality and non-spectrality in a simple manner. Apply-ing the given method, we clarify the spectrality and non-spectrality of a class of self-affine measures supporting on the generalized spatial Sierpinski gasket （that is, M is an upper（lower）-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）.In the second part, we deal with the case that M= diag[p1,P2,P3]∈Mn（Z） and D ={0,e₁+e₂,e₁+e₃,e₂+e₃}. The main result says that μM,D is spectral measure if pj∈2Z\{0}（j=1,2,3） or Pj ∈ 2Z\{0}（j=1,2）, p3∈2Z+1\{±1}. This generalizes the known result. Also it is equivalent to the statement that if M ∈ M₃（R） a special matrix and D={0, e₁,e₂, e₃}, then μM,D is spectral measure. Such a special matrix M ∈ M₃（R） may have entries in Q, which extends the range of the entries of M from Z to Q.