Non-Spectrality of Certain Self-Affine Measures on the Generalized Spatial Sierpinski Gasket

Let μM,Dbe a self-affine measure associated with an expanding integer matrix M=[ p1,0,0;p4,p2,0;p5,0,p3]and the digit set D={ 0,e1,e2,e3}in the space R3, where p1,p2,p3∈Z\{ 0,±1 }, p4,p5∈Zand e1,e2,e3are the standard basis of unit column vectors in R3. In this paper, we mainly consider the case p1,p2,p3∈2Z+1, p2≠p3, p4=l(p1−p2), p5=l(p3−p1),where l∈2Z. We prove that μM,Dis a non-spectral measure, and there are at most 4-element μM,D-orthogonal exponentials, and the number 4 is the best. The results here generalize the known results.

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self- affine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-aiffine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.

A necessary and sufficient condition for the finite μM,D-orthogonality

The self-affine measure μM,D associated with an expanding matrix M ∈ Mn(Z) and a finite digit set D ? Znis uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πiλ,x : λ∈Λ} in the Hilbert space L2(μM,D) is simply called μM,D-orthogonal exponentials. We consider in this paper the finiteness of μM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite μM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.

Analysis of a class of spectral pair conditions

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.

Digit Sets for Self-similar Tiles

A new class of digit sets, which we called very weak product-form digit sets, is introduced and it is shown that they are tile digit set for self-similar tiles. This extends previous results about product-form and weak product-form digit sets.