Spectrality Of Self-affine Measures For A Class Of Special Digit Sets And Direct Sum Digit Sets

The self-affine measure associated with an expanding and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self-affine measures have been received much attention in recent years. This paper mainly discusses the spectral problems of self-affine measures determined by a class of digit sets and expanding integer matrix by drawing references from the research results of predecessors, and then we shall extend the2direct sum digit sets into3, and finally some conclusions about spectral and non-spectral self-affine measures are given.The main results of the paper are as follows:The first part obtains the self-affine measures determined by digit set and integer diagonal form or upper (lower) triangular matrices are spectral measures base on one of Strichartz’es theorems. In which, this kind of digit set is the extension of previous digit sets that is direct sum digit set. The improvement is much certain significant.In the second part, we extend the spectrality and non-spectrality of planar self-affine measures with two-element digit sets into high dimensional space simply.And in the third part, we shall extend the spectrality and non-spectrality of self-affine measures with2direct sum digit sets into3direct sums. If the Fourier transform of self-affine measure satisfies some conditions, then the self-affine measures are spectral measures. Similarly, if the decomposable digit sets have some special characteristics, then the self-affine measure is non-spectral measure.