Spectrality of planar self-affine measures with two-element digit set

The iterated function system with two-element digit set is the simplest case and the most important case in the study of self-affine measures. The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable. The higher dimensional analogue is not known, for which two conjectures about the spectrality and the non-spectrality remain open. In the present paper, we consider the spectrality and non-spectrality of planar self-affine measures with two-element digit set. We give a method to deal with the two-dimensional case, and clarify the spectrality and non-spectrality of a class of planar self-affine measures. The result here provides some supportive evidence to the two related conjectures.