## Analysis on self-similar sets

##### Author(s)

##### Advisor

Gheondea E., Aurelian B. N.##### Date

2020-08##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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##### 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.

##### Keywords

Self-similar setShift space

Shift map

Critical set

Post-critical set

Self-similar structure

Post-critically ﬁnite set