Applications of compact topological graph c*-algebras to noncommutative solenoids

We study the KMS states of the Toeplitz extension of the noncommutative solenoids introduced by Latremoliere and Packer. We demonstrate that noncommutative solenoids cannot be constructed as a direct limit of C*-algebras arising from inverse sequences of topological graphs associated to local homeomorphisms of T. We employ a different approach, utilizing a topological analogue of higher power graphs and Katsura's factor maps to obtain a noncommutative solenoid as a direct limit of topological graph algebras. This approach is compatible with the Toeplitz algebra of topological graphs, and enables us to define a Toeplitz extension of each noncommutative solenoid. We expand on the results of KMS states of finite-graph Toeplitz-algebras, developing analagous results for the Toeplitz algebras of compact topological graphs. This is done with the aim of understanding the KMS states of noncommutative solenoids. The final chapter of the thesis deals with an attempt to characterise the KMS states of Toeplitz noncommutative solenoids, under a positivity assumption. We conclude with the conjecture that these KMS states are unique for β > 0.