Some Results on Planar Self-Affine Measures with Collinear Digit Sets

Ever since Jorgensen and Pedersen (J Anal Math 75:185-228, 1998) discovered the first singular spectral measure, the spectral and non-spectral problems of fractal measures have received a lot of attention in recent years. In this work, we study the planar self-affine measure mu(M, D) generated by an expanding matrix M is an element of M-2(Z) and a collinear digit set D = {0, d(1), d(2), d(3)}v, where v is an element of Z(2)\{0} and d(1), d(2), d(3) are different non-zero integers. For the case that {v, Mv} is linearly dependent, the sufficient and necessary condition for mu(M, D) to be a spectral measure is given. Moreover, we estimate the number of orthogonal exponential functions in L-2(mu(M, D)) and give the exact maximal cardinality when mu(M,D) is a non-spectral measure. At the same time, partial results are also obtained for the case that {v, Mv} is linearly independent.