## Honors Scholar Theses

#### Date of Completion

Spring 5-1-2022

Vasileois Chousionis

Mathematics

#### Disciplines

Analysis | Dynamical Systems | Mathematics | Physical Sciences and Mathematics

#### Abstract

This thesis is an expository investigation of the conformal iterated function system (CIFS) approach to fractals and their dimension theory. Conformal maps distort regions, subject to certain constraints, in a controlled way. Let $\mathcal{S} = (X, E, \{\phi_e\}_{e \in E})$ be an iterated function system where $X$ is a compact metric space, $E$ is a countable index set, and $\{\phi_e\}_{e \in E}$ is a family of injective and uniformly contracting maps. If the family of maps $\{\phi_e\}_{e \in E}$ is also conformal and satisfies the open set condition, then the distortion properties of conformal maps can be extended to the system $\mathcal{S}.$ The behavior of the system can be modeled via thermodynamic formalism, which introduces notions such as the topological pressure and the Perron-Frobenius operator. Both are critical to developing numerical approximations for the dimension of the limit set of the system. Finally, we provide examples of fractals which are well-described by the CIFS framework.

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