Non-spectrality And Singularity Of Some Self-affine Measures With Expanding Real Matrix

How to determine the non-spectrality and singularity of self-affine measure have been received much attention in the past years. Based on the research results of predecessors, this paper mainly research on the number of orthogonal exponentials in L2(μM,D)-sPace which is decided by the expanding real matrix and finite digit sets, and the singularity of self-affine measure determined by some expanding matrix which contain Pisot number and some special digit sets.The main results are as follows:In the first part, we extend the research results of P. E. T. Jorgensen and J.-L. Li which given way of judging the non-spectrality of self-affine measure associated with M∈Mn(Z), D(?)Zn. By analyzing the character of the zeo set o μM,D, we obtain some sufficient conditions to judge whether exist finite orthogonal exponentials or not in L2(μM,D)-space when M∈Mn(R), D(?)Zn. At the same time, we give some examples as the applications for every main result.In the second part, we obtain some sufficient conditions to judge the singularity of self-affine measure associated with some special expanding matrixes with Pisot number and some generalized digit sets by using Rieman-Lebesgue Lemma when there exists a positive lower bound for the Fourier transformation of self-affine mea-sure. At the same time, these results popularize the related conclusions obtained by P. E. T. Jorgensen, K. A. Kornelson, K. L. Shuman and L.-S. Zhang.