The Spectrality of Self-affine Measure Under the Similar Transformation of GLn(p)

Let μ M , D be the self-affine measure generated by an expanding integer matrix M ∈ M n ( Z ) and a finite digit set D ⊂ Z n . It is well known that the two measures μ M , D and μ M ~ , D ~ have the same spectrality if M ~ = B - 1 M B and D ~ = B - 1 D , where B ∈ M n ( R ) is a nonsingular matrix. This fact is usually used to simplify the digit set D or the expanding matrix M . However, it often transforms integer digit set D or integer expanding matrix M into real, which brings many difficulties to study the spectrality of μ M ~ , D ~ . In this paper, we introduce a similar transformation of general linear group G L n ( p ) for some self-affine measures, and discuss their spectral properties. This kind of similar transformation can keep the integer properties of D and M simultaneously, which leads to many advantages in discussing the spectrality of self-affine measures. As an application, we extend some well-known spectral self-affine measures to more general forms.