Spectrality of certain self-affine measures on the generalized spatial Sierpinski gasket

The self-affine measure mu(M,D) corresponding to a upper or lower triangle expanding matrix M and the digit set D = {0, e(1), e(2), e(3)} in the space R-3 is supported on the generalized spatial Sierpinski gasket, where e(1), e(2), e(3) are the standard basis of unit column vectors in R-3. We consider in this paper the existence of orthogonal exponentials on the Hilbert space L-2(mu(M,D)), i.e., the spectrality of mu(M,D). Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non-spectrality of mu(M,D). As an application, the spectral property of a class of such self-affine measures are clarified. The results here generalize the corresponding results in a simple manner. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim