On spectral and non-spectral problem for the planar self-similar measures with four element digit sets

作者：Li, Qian; Wu, Zhi-Yi^*

来源：Forum Mathematicum, 2021, 33(6): 1629-1639.

DOI：10.1515/forum-2021-0173

• Infinite orthogonal set
• spectral measure
• orthogonal exponential functions
• Fourier transform

We consider the self-similar measure mu(M,D) generated by an expanding real matrix @@@ M = (rho(-1) 0 0 rho(-1)) is an element of M-2(R) @@@ and a digit set @@@ D = {(0 0), (a b), (c d), (a + c b + d)} subset of Z(2). @@@ In this paper, we study the spectral and non-spectral problems of mu(M,D). In this case that (a b) and (c d) are two independent vectors, we prove that if rho(-1) is an element of Z, then mu(M,D) is a spectral measure if and only if rho(-1) is an element of 2Z. For the case that (a b) and (c d) are two dependent vectors, we first give the sufficient and necessary condition for L-2(mu(M,D)) to contain an infinite orthogonal set of exponential functions. Based on this result, we can give the exact cardinality of orthogonal exponential functions in L-2(mu(M,D)) when L-2(mu(M,D)) does not admit any infinite orthogonal set of exponential functions by classifying the values of rho.

• 单位
  广州大学