Analysis on self-similar sets

Abstract

Self-similar sets are one class of fractals that are invariant under geometric similarities. In this thesis, we study on self-similar sets. We give the definition of a self-similar set K and present the proof the existence theorem of such a set. We define the shift space. We define a relation between the shift space and K. We show the self-similarity of the shift space. We define overlapping set, critical and post-critical set for a self-similar set. We give the characterization of K by the periodic sequences in the shift space. We give the notion of a self-similar structure and define a self-similar set purely topologically. We give its local topology. We define isomorphism between selfsimilar structures so that we can have a classification of self-similar structures. We point out that the critical set for a self-similar structure provides us with a characterization for determining the topological structure of a self-similar structure. We define the notion of minimality for a self-similar structure and give a characterization theorem for investigating the minimality of a self-similar structure. We define a post-critically finite self-similar structure.