ON THE FOURIER ORTHONORMAL BASES OF A CLASS OF MORAN MEASURES ON R2

Let mu{R-i},{D-i} be the probability measure generated by the iterated function system (IFS): {F-Ri,D-i (x) = R-i(-1) (x + d) : d is an element of D-i}(i=1)(infinity), where R-i = rho (GRPHICS) is an expanding matrix with 1 < rho is an element of R, d(i) is an element of Z, and D-i = {(GRPHICS), (GRPHICS), (GRPHICS)} with k(i) is an element of Z, and sup(i is an element of N) {vertical bar d(i)vertical bar, vertical bar k(i)vertical bar) < infinity. In this paper, we consider the spectral properties of mu {R-i},{D-i}, we show that mu{R-i},(D-i).} is a spectral measure, i.e., there exists a countable set Lambda subset of R-2, such that E(Lambda) := {e(2 pi i(x,lambda)), lambda is an element of Lambda} forms an orthonormal basis for L-2 (mu({Ri},{D)i(})), if and only if rho = 3k for some k is an element of N. Furthermore, we also provide an equivalent characterization for the maximal bi-zero set for mu({Ri},{Di}) by defining a mixed tree mapping for it. And we also obtain some results associated with the Sierpinski-type measures.