Two Problems in Graph Algebras and Dynamical Systems

There are two parts to this dissertation. The first topic comprises Chapter two of this document, where we consider classification of nonsimple graph C∗-algebras. There are many classes of nonsimple graph C∗-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C∗-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C∗-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C∗-algebras.

The second part considers a problem in dynamical systems. We prove that Lyapunov exponents of infinite-dimensional dynamical systems can be computed from observational data. Crucially, our hypotheses are placed on the observations, rather than on the underlying infinite-dimensional system. We formulate checkable conditions under which a Lyapunov exponent computed from experimental data is a Lyapunov exponent of the underlying infinite-dimensional dynamical system (provided that the observational scheme is typical in the sense of prevalence).