Integral Self-afne Tiles And Asymptotic Properties Of Substitution

Integral self-a?ne tiles and sequences of substitutions are both important and in-teresting in fractal geometry. In this paper, several results on the integral self-a?netiles of Bandt's model and the asymptotic properties of unoriented walk of substitutionare obtained.The first part deals with integral self-a?ne tiling of Bandt's model. This is ageneralization of the integral self-a?ne tiling. The invariant set of the integral self-a?ne tiling system tiles the space by translations. While we use the invariant set ofBandt's model to tile the space, rotations and re?ections must be allowed. Indeed,Bandt's model gives us many interesting tiles such as Levy dragon, But it receives veryfew studies until now. Based on the work of Bandt, Keyon, Lagarias and Wang, usingergodic theory, we show that the Lebesgue measure of the tile is a rational numberwhere the denominator equals the order of the associate symmetry group. We applythe result to the study of Levy dragon, and with the help of disjoint neighbor relationtechnique, we firstly get that the Lebesgue measure of Levy dragon is 1/4.The second part deals with asymptotic properties of unoriented walk of substi-tution. The substitutions sequence starts from original point, walks up one if it meet"a"and walks down one if it meet"b", which is called the unoriented walk of thesequence(the step length is 1). J.Peyrie′re gave a problem: What's the asymptotic prop-erty of this kind of walk? That is, is the unoriented walk bounded, tending infinite, orrecurrent? Zhi-Xiong Wen and Zhi-Ying Wen give the answer by using the maximalalgebra and the minimal algebra they defined. The step length is one in their paper;In this paper we obtain the asymptotic property with any step length. We hope to dealwith the asymptotic proerty of Rauzy sequence, which has three alphabets.In conclusion, this dissertation brings the following new ideas.We prove that the Lebesgue measure of the Bandt's tile is a rational number byusing ergodic theory. We firstly give the exact Lebesgue measure of Levy dragon.We generalize the technique of maximal algebra and minimal algebra to the un-oriented walk with any step length.