Dimension Theory of Conformal Iterated Function Systems

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