Spectrality Of The Planar Sierpinski Family

Let A∈M2（Z2） be an expanding matrix and let D∈Z2 be a digit set as follows: D={（0,0）*, （1,0）*, （0,1）*}, where v* means the compose of the vector v. Then there exists a unique probability measure μA,D satisfying that which is called a Sierpinski measure.In this thesis we study the spectrality of Sierpinski measures, that is, to inves-tigate the existence of a countable set Λ such that E（Λ）={e^{-2πi（λ,x）}:λ∈Λ} is an orthonormal basis （Fourier basis） for L²(（μA,D）. We prove that the Sierpinski measure μA,D is a spectral one if and only if （A, D） is admissible. This answer the open problem raised by Li （Sci. China Math.,56 （2013）,1619-1628）. Combining the series works of Li on this issue, we obtain the characteristic conditions for the spectrality of Sierpinski measures. In the same time, we give a positive answer for the following open problem:Under what conditions on an integral matrix A and digit set C（？）Z², the integral self-affine measure μ_A,C is a spectral one if and only if （A, C） is admissible. That is, we show that the open problem is true for Sierpinski measures.