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

We model hyperbolic toral automorphisms by two types of Smale spaces; shifts of finite type and substitution tilings spaces. Smale spaces are dynamical systems with local hyperbolic product structure. In 1970, Bowen showed that an irreducible Smale space is a factor of a shift of finite type by showing that it has Markov partitions. Putnam extended Bowen's theorem by showing that every irreducible Smale space has a factor map that can be split into a s-bijective and u-bijective map; thereby better modelling a Smale space on its characterizing expanding and contracting spaces separately. In this thesis, we define two new constructions of Markov partitions for hyperbolic toral automorphisms inspired by the work of Adler, Weiss, and Praggastis. With one of the constructions, we investigate when a factor map from a shift of finite type to a hyperbolic toral automorphism can be written as a composition of a s-bijective and u-bijective map and we show that if such a splitting exists then the Markov partition must satisfy a Border Continuity condition. The second construction can be thought of as an explicit example of Putnam's theorem for the case of hyperbolic toral automorphisms whose defining matrix is in dimension 2 and has positive entries. We define a full splitting for all such hyperbolic toral automorphisms with one exception; the Arnold Cat map. / Graduate