Spectral structure of Moran Sierpinski-type measure on R2

Let M-n = diag [3p(n), 3q(n)] with p(n), q(n )>= 1 for all n >= 1 and let D = {(0, 0)(t),(1, 0)(t),(0, 1)(t)}. One can generate a Borel probability measure mu{M-n}, D = delta D--1(M1) & lowast; delta D--1((M2M1)) & lowast;delta D--1((M3M2M1)) & lowast;... . Such measure mu({Mn},D) is called a Moran Sierpinski-type measure. It is known Denget al(Acta Math. Sin. submitted) that the associated Hilbert space L-2(mu({Mn},D)) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L-2(mu({Mn},D)). For sucha maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L-2(mu({Mn},D))or not.