The Connectedness Of Planar Self-affine Set And The Tile Digit Set

In this thesis, we study the connectedness of self-affine set T(A, D) gen-erated by an expanding matrix and a product digit set D= {0,1,...,m-1}× {0,1,...,n-1}, wherep,q ∈Z,m,n∈Z+,a ∈R. A nec-essary and sufficient condition for T(A, D) to be connected is given.Moreover, we discuss the conditions so that T(A,D) to be a tile. There are four chapters in this thesis.In chapter 1, we introduce the background, the current status and the significance of the research on self-affine sets and self-affine tiles and states the main results of this thesis.In chapter 2, we introduce some basic knowledge and lemma associated with this thesis, including criterion for connectedness of self-affine sets.In chapter 3, we study the connectedness of T(A,D) under condition |p|+1< m< 2|p| - 1, giving a necessary and sufficient condition for T(A, D) to be connected.In chapter 4, we discuss the tiling property of T(A, D), giving the rela-tionship between m,n, a when T(A,D) to be tile.