The Research On The Connectedness Of Two-dimensional Self-affine Sets

Recently, the class of self-affine tiles has attracted a lot of attention due to its special roles in fractal geometry, wavelet theory and the spectral set problem of Fuglede. Despite the extensive studies, there are still many aspects of the self-affine tiles that are not fully understood. In particular, there is very limited knowledge on the basic topological properties such as the connectedness or the disklikeness (homeomorphic to the closed unit disc). In the thesis, we will mainly study the connectedness of some kinds of self-affine sets, one with its digit set lying in a parallelogram and the other with its digit set lying in a rectangle. We find out the sufficient and necessary condition for the connectedness of the first kind, as to the latter, the sufficient condition is given. We introduce some basic knowledge associated with fractal geometry and the well-developed theory on the connectedness of self-affine sets in Chapter 1 and Chapter 2. The core of this paper is in Chapter 3 and Chapter 4.In Chapter 3, we investigate the connectedness of the self-affine sets gener-ated by matrix and digit set where p, q∈Z,3≤|p|+1≤|q|<2|p|-1 and D1={0,s,…(|q|-1)s},D2= {0, t,…(|p|-1)t}, s, t≠0, and find out the sufficient conditions.In Chapter 4, we investigate the connectedness of the self-affine sets gen-erated by matrix and digit set' |p|-1,0≤j≤|q|-1.}, where s, t≠0, p, q, d∈Z,|p|,|q|≥2 and reveal the sufficient and necessary conditions.