Two Problems in Graph Algebras and Dynamical Systems



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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).



Range, K-theory