THE SPECTRAL EIGENMATRIX PROBLEMS OF PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

Given a Borel probability measure mu on R-n and a real matrix R is an element of M-n(R). We call R a spectral eigenmatrix of the measure mu if there exists a countable set Lambda subset of R-n such that the sets E-Lambda = {e(2 pi i <lambda,x >) : lambda is an element of Lambda} and E-R Lambda = {e(2 pi i < R lambda, x >) : lambda is an element of Lambda} are both orthonormal bases for the Hilbert space L-2(mu). In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure mu(M,D) generated by an expanding integer matrix M is an element of M-2(2Z) and the four-elements digit set D = {(0, 0)(t), (1, 0)(t), (0, 1)(t), (-1, -1)(t)}. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of mu(M,D) are given.