Spectrality of self-affine Sierpinski-type measures on R2

Authors：Dai, Xin-Rong; Fu, Xiao-Ye; Yan, Zhi-Hui^*

Source：Applied and Computational Harmonic Analysis, 2021, 52: 63-81.

DOI：10.1016/j.acha.2019.12.001

• Spectral measure
• Self-affine measure
• Bi-zero set
• Spectrum

In this paper, we study the spectral property of a class of self-affine measures mu(R,D) on R-2 generated by the iterated function system {phi(d) (.) = R-1 (.+d)}(d is an element of D) associated with the real expanding matrix R= (graphics) and the digit set D = {((0)(0)), ((1)(0)), ((0)(1))}. We show that mu(R,D) is a spectral measure if and only if 3 vertical bar b(i, )i = 1, 2. This extends the result of Deng and Lau [J. Funct. Anal., 2015], where they considered the case b(1) = b(2). And we also give a tree structure for any spectrum of mu(R,D) by providing a decomposition property on it.

• Institution
  中山大学