Spectrality of homogeneous Moran measures on the plane

Abstract

LetM=ρ1−1a0ρ2−1 be a real upper triangular expanding matrix and D=00,d10,d2d3 be a three-element real digit set with d1d3≠0 , and let {nk}k=1∞ be a sequence of positive integers with upper bound. The infinite convolutions μM,D,{nk}=δM−n1D∗δM−(n1+n2)D∗⋯∗δM−(n1+⋯+nk)D∗⋯ converges weakly to a Borel probability measure (homogeneous Moran measure). In this paper, we study the existence of exponential orthonormal basis for L2(μM,D,{nk}). A necessary and sufficient condition for μM,D,{nk} to be a spectral measure is established.