Analysis of μM,D-orthogonal exponentials for the planar four-element digit sets

The self-affine measure mu(M,D) is a unique probability measure satisfying the self-affine identity with equal weight. It only depends upon an expanding matrix M and a finite digit set D. In this paper we study the question of when the L-2(mu(M,D))-space has infinite families of orthogonal exponentials. Such research is necessary to further understanding the spectrality of mu(M,D). For a class of planar four-element digit sets, we present several methods to deal with this question. The application of each method is also given, which extends the known results in a simple manner. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim