The Properties Of Linear Iterated Function Systems

This thesis mainly introduces the linear iterated function system with overlaps and focuses on apply these result to the square case.This article centers on the paper, which refers to Combinatorics of linear iterated function systems with overlaps(Sidorov published in year 2007).The main body can be divided into two parts.Part One:We mainly introduce the theorem in Sidorov's[31 paper.We first introduce the linear iterated function system by two examples.Then we focuses on the properties of a pair of similitudes f0(x)=?x and f1(x)=?x+1-?,and the relationship between these properties and the parameter ? while ??(0,1) and x?I=[0,1].If ??(0,1/2],the unique self-similar attractor S?(I)=f0(x)?f1(x) satisfies the open set condition(OSC). But if ??(1/2,1),S?(I)does not satisfy the OSC.Erdos,Joo and Komornik proved the theorem which shows that if ?>g=(?5-1)/2?0.618, has the cardinality of the continuum for each x?(0,1).Sidorov generalizes the result of above theorem to the multidimensional case.Let p0,p1,…,pm-1?Rd,and {fj}j=0m-1 is the one-parameter family of similitudes in Rd,and fj(x)=?x+(1-?)pj(j=0,1,…,m-1), where the parameter ??(0,1).Sidorov proved a result as following:for each p0,p1,…,pm-1, there exists ?0<1,such that when ??(?0,1),S?=? and each x??\{p0,…,pm-1} has 2N0 distinct addresses.Part Two:we consider the square case and obtain the following result: Let ?0=(?5-1)/2,which is the unique positive root of ?2+?=1.Then we prove that where ?<?0,there exists x??\{p0,…,pm-1} has unique address and when ??(?0,1), each x??\{p0,…,pm-1} has 2(?)0 distinct addresses.