The Noncommutative Dynamics and Topology of Iterated Function Systems

Using operator algebraic techniques, we explore the relationship between the dynamics and topology of iterated function systems. We examine the C*-algebras introduced by Kajiwara and Watatani, and extend many of their results to the non-contractive setting with large overlaps. We build upon their work with invertible systems to show that every Kajiwara-Watatani algebra is a subalgebra of an Exel crossed product. An investigation into the feasibility of a groupoid model for Kajiwara-Watatani algebras is undertaken, and in the process we develop novel topological techniques for groupoid C*-algebras. Finally, we introduce a new C*-algebra, called the lacunary algebra, which is built from an iterated function system. This algebra is sensitive to the interaction between the topology and dynamics of the system, and we compute its K-theory for an illustrative class of examples.