The Connectedness Of Self-affine Set And Spectral Properties Of Self-affine Measure

Let M∈Mn(Z) is an expanding integer matrix and D ∈ Zn is a finite digit set. Then the family of maps{φd(x)= M-1(x+d)}d∈D are contractive under a suitable norm in Rn, and it is well known that there is a unique nonempty compact set T=T(M,D) satisfying the set equation T=∪d∈Dφd(T). T is called the invariant set (or attractor) of the iterated function system (IFS) ｛φd}d∈D.Each invariant set defined above is naturally equipped with a unique in-variant probability measure μ:=μM,D, the invariance defined from the given affine system{φd}d∈D by This is a self-affine identity with equal weight, and μM,D is supported on T(M,D). The invariant measure μM,D is also called self-affine measure.This article consists of three chapters, among which we study the connect-edness of a class of planar self-affine set in the first half of this article, as well as the spectral properties of a class of higher dimensional self-affine measure in the second half.In the first Chapter, the background and current situation of the connect-edness of self-affine sets and the spectral properties of self-affine measures are introduced. At the same time, the main results of this article are given.In Chapter 2, we consider the connectedness of two class of planar self-affine sets T(M,D):(1)Let the characteristic polynomial of the expanding integer matrix M be f(x)= x~2+bx+c and digit set D={0,1,...,m}v, the necessary and sufficient conditions only depending on the integers m, b, c are given for the T(M,D) to be connected. (2)Let the characteristic polynomial of the expanding integer matrix M be f(x)=x~2-(p+q)x+ pq and digit set D={0,1,...,|pq|-2,|pq|-1+s}v, the necessary and sufficient conditions only depending on the integers s,p, q are given for the T(M,D) to be connected.In Chapter 3, We study the spectral properties of higher dimensional self- affine measure μM,D generated by a expanding integer matrix M ∈ Mn(Z) together with consecutive and collinear digit set D={0,1,…,q-1}v. the main results are as follows:(1)when{v, Mv, M2v,…,Mn-1v} is linear-ly independent, if gcd(q, det(M))≠1, then μM,D has infinitely many or-thogonal exponentials; if q|det(M), then μM,D is a spectral measure. (2)If {v,Mv,Mv,…,Mn-1v} is linearly dependent, we can transform it into (1) by reduce the dimension, thus, the sufficient conditions for μM,D to be a spec-tral measure or has infinitely many orthogonal exponentials also can be got.