A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is completely determined by a C*-correspondence, which consists of a right Hilbert A- module, E, and a *-homomorphism from the C*-algebra A into L(E), the adjointable operators on E. Some familiar examples of C*-algebras which can be recognized as Cuntz-Pimsner algebras include the Cuntz algebras, Cuntz-Krieger algebras, and crossed products of a C*-algebra by an action of the integers by automorphisms. In this dissertation, we construct a Cuntz-Pimsner Algebra associated to a dynam- ical system of a substitution tiling, which provides an alternate construction to the groupoid approach found in [3], and has the advantage of yielding a method for com- puting the K-Theory. / Graduate

There is a long history of interesting connections between topological dynamical systems and C*-algebras. Iterated function systems are an important topic in dynamics, but the diversity of these systems makes it challenging to develop an associated class of C*-algebras. Kajiwara and Watatani were the first to construct a C*-algebra from an iterated function system. They used an algebraic approach involving Cuntz-Pimsner algebras; however, when investigating properties such as ideal structure, they needed to assume that the functions in the system are the inverse branches of a continuous map. This excludes many famous examples, such as the standard functions used to construct the Siérpinski Gasket. In this thesis, we provide a construction of an inductive limit of étale equivalence relations for a broad class of affine iterated function systems, including the Siérpinski Gasket and its relatives, and consider the associated C*-algebras. This approach provides a more dynamical perspective, leading to interesting results that emphasize how properties of the dynamics appear in the C*-algebras. In particular, we show that the C*-algebras are isomorphic for conjugate systems, and find ideals related to the open set condition. In the case of the Siérpinski Gasket, we find explicit isomorphisms to subalgebras of the continuous functions from the attractor to a matrix algebra. Finally, we consider the K-theory of the inductive limit of these algebras. / Graduate