THE NON-SPECTRAL PROPERTY OF A CLASS OF PLANAR SELF-SIMILAR MEASURES

Let the self-similar measure mu M, D be generated by an expanding real matrix M = rho I-1 is an element of M-2(R) and a digit set D = {(0, 0)(t), (1, 0)(t), (0, 1)(t), (-1,-1)(t)} in space R-2. In this paper, we only consider rho > 0 and the case rho < 0 is similar. We show that there exists an infinite orthogonal set of exponential functions in L-2(mu M,D) if and only if rho = (q/(2p))(1/r) for some p, q, r is an element of N with gcd(2p, q) = 1. Furthermore, for the cases that L-2(mu M,D) does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in L-2(mu M,D) is given.