Orthogonal Exponentials of Planar Self-Affine Measures with Four-Element Digit Set

Abstract

. Let µ M , D be a self-affine measure generated by an expanding real matrix M = (cid:18) a e f b (cid:19) and the digit set D = { ( 0,0 ) t , ( 1,0 ) t , ( 0,1 ) t , ( 1,1 ) t } . In this paper, we con-sider that when does L 2 ( µ M , D ) admit an infinite orthogonal set of exponential functions? Moreover, we obtain that if e = f = 0 and a , b ∈{ pq , p , q ∈ 2 Z + 1 } , then there exist at most 4 mutually orthogonal exponential functions in L 2 ( µ M , D ) , and the number 4 is the best possible.