Some Studies Of Two Problems Of The Self-Similar Sets

The self-similar sets are the most important class in Fractals, which has beenstudied extensively and deeply. In this Thesis, we study two problems of self-similarsets: similarity of the union of finite intervals, and properties of the self-similar-likeset.The first part deals with similarity of the union of finite intervals. One of the basicproblem is how to construct new self-similar set from some known self-similar sets, atypical question is if the union of two closed intervals is still a self-similar set satisfyingthe open set condition, and secondly whether the union of any finite closed intervalsis still a self-similar set satisfying the open set condition. In this paper we will definethe multiple word and incidence matrices corresponding to the union of finite intervals.We will show the relation between the irreducibility of multiple word and incidencematrices. Then from properties of its multiple word and incidence matrices, we willcharacterize the similarity fulfilling open set condition of the union by the lengths ofthe intervals and the gaps between consecutive intervals.The second part deals with properties of the self-similar-like set. The classicalself-similar set (SSS) is a compact set, but in many cases, for example in Multifrac-tal, we have to treat non-compact sets with self-similar structure, we call those non-compact sets the self-similar-like sets (SSLS). In this Thesis we will show that if theIFS satisfies the strong separate condition, for any point in the self-similar set, its mixedorbit set is a minimal self-similar-like set. From this we will characterize the structureof self-similar-like sets, and give some basic properties of them. We will show that theclass of all self-similar-like set is closed under the operations”union”,”intersection”,”complementary”. We also give the definition and some properties of the generating setof self-similar-like sets, we will show that the Hausdorf dimension of a self-similar-like set is equal to any one of its generating sets, which give a method to estimate theHausdorf dimension of a self-similar set or self-similar-like set. In the general cases, we will generalize the definition of the generating set of self-similar-like sets, and wewill show that the Hausdorf dimension of a self-similar-like set is also equal to anyone of its generating sets for each IFS.In conclusion, this dissertation brings the following new ideas: we characterizethe similarity fulfilling open set condition of the union by the lengths of the intervalsand the gaps between consecutive intervals. We define the self-similar-like sets, char-acterize the structure of them, and give their basic properties; we give the definitionof the generating set of self-similar-like sets, show that the Hausdorf dimension of aself-similar-like set is equal to any one of its generating sets. sets.