Two-dimensional Self-affine Tiles With Nonconsecutive And Noncolinear Digit Sets

A∈Mn(R) is an expanding matrix, that is, all its eigenvalues have mod-ulus>1. m=|det A|is some integer, and D={d1,..., dm}∈Rn is a digit set of m distinct vectors. Then, the self-affine set T(A, D) generated by A and D can be explicitly given by T(A,D)=∑k=1∞A-kdk:dk∈D}. T(A,D) is called a self-affine tile if it has positive Lebesgue measure. T(A,D) is called an integral self-affine tile if A∈Mn(Z) and D∈Zn.In this thesis, we study self affine set generated by a lower triangular expanding matrix A and a noncolinear digit set D. There are four chapters in this thesis and the main contents and results are summaried as follows:In Chapter1, we introduce the background, the significance of the research on fractals, self-affine sets and self-affine tiles, and state the main results.In Chapter2, we introduce some basic known results associated with this thesis, including criterion for a self-affine-set to be a self-affine tile, and the well-developed theory on the connectedness and disklikeness (homeomorphic to the closed unit disc) of self-affine set.In Chapter3, we study self-affine tiles generated by lower triangular ex-panding matrices and consecutive and noncolinear digit sets. We obtain nec-essary and sufficient conditions for the self-affine tile to be connected and disklike, and the tiling set.In Chapter4, we study the self-affine sets generated by lower triangular expanding matrices and nonconsecutive and noncolinear digit sets. We obtain a sufficient condition for a self-affine set to be a self-affine tile. Furthermore, we study a special class of self-affine sets generated by lower triangular expanding matrices and nonconsecutive and noncolinear digit sets. We can prove that the self affine set is a self affine tile, and obtain necessary and sufficient conditions for the self-affine tile to be connected and disklike, and the tiling set.