Circle Maps and C*-algebras

Let $\T$ be the unit circle in the complex plane, and let $\phi:\T\to\T$ be a map which is continuous, surjective and piecewise monotone. We stress that $\phi$ is allowed to have critical points. This thesis introduces a construction of a two \'etale groupoids, $\Gamma_\phi$, $\Gamma_\phi^+$, from such a map, generalising the transformation groupoid of a local homeomorphism first introduced by Renault in \cite{re}. We conduct a detailed study of the relationship between the dynamics of $\phi$, the properties of these groupoids, the structure of their corresponding reduced groupoid $C^*$-algebras, and, for certain classes of maps, the K-theory of these algebras. When the map $\phi$ is transitive, we show that the algebras $C^*_r(\Gamma_\phi)$ and $C^*_r(\Gamma_\phi^+)$ are purely infinite and satisfy the Universal Coefficient Theorem. Furthermore, we find necessary and sufficient conditions for simplicity of these algebras in terms of dynamical properties of $\phi$. We proceed to consider the situation when the algebras are non-simple, and describe the primitive ideal spectrum in this case. We prove that any irreducible representation factors through the $C^*$-algebra of the reduction of the groupoid to the orbit $[x]$ of some point $x\in\T$, and the corresponding primitive ideals come in two distinct types, determined by the isotropy over this point $x$. Next, we study critically finite maps -- maps for which the forward orbit of any critical point is finite -- and develop an algorithm for calculating for calculating the K-theory groups of the groupoid $C^*$-algebras of such a map. Finally, we study continuous, piecewise monotone circle maps without periodic points. We show that the groupoid $C^*$-algebras corresponding to such maps have a unique maximal ideal, and then use results by Putnam, Skau and Schmidt to determine the K-theory of these $C^*$-algebras.