Self-affine Sets With Positive Lebesgue Measures

In this paper,we mainly study the problem about self-affine regions, that is, self-affine sets with positive Lebesgue measure. There are two parts. In the first part,by depicting the characteristic of digit set D,we obtain a sufficient condition for it to give rise to self-affine regions. In the second part,we depict the characteristic of digit set D of integral self-affine tiles with prime determinant, which is a special kind of self-affine regions.Usually, we are difficult to find the attractor of iterated function systems, so it is difficult to calculate its Lebesgue measureμ(T).And we have already known that for most pair (M, D),it has Lebesgue measureμ(T)=0.Therefore which we take more focus on are,under what conditions, for a given matrix M and diget set D, there isμ(T)>0 and its inverse,that is, ifμ(T)>0, then what properties do the matrix M and diget set D must have? These are important subjects which have been investgated in recent years.The main findings are as follows:1.Obtaining a sufficient and necessary condition for self-affine sets being self-affine regions and following it,a sufficient condition for digit sets to give rise to self-affine regions is obtained. That is, if D is a quasi-product form digit set,then T(M, D) is a self-affine region. This simplifies the proof of the corresponding result of document[1].2.Discussing the characteristic of digit sets of integral self-affine tiles with prime determinant,that is,if T(M,D) is a self-affine tile, with|det(M)|= p and pZn(?)M2(Zn),then we have the following results:(ⅰ)Z[M, D](?)M(Zn)(?)D is a complete comset representation of Zn/M(Zn).(ⅱ)Z[M,D](?)M(Zn)(?)D=MγD0,withγ∈N*,D0 is a complete comset representation of Zn/M(Zn).Note that the condition pZn(?)M2(Zn) in the theorem isn't necessary, which could be weakened as span(D)=Rn, And furthermore, it also can obtain the cor-responding results without any other additional assumption in the two-dimensional case.