This thesis describes the equilibrium states (the KMS states) of dynamical systems arising from local homeomorphisms. It has two main components. First, we consider alocal homeomorphism on a compactspace and the associated Hilbert bimodule.This Hilbert bimodule has both a Toeplitz algebraand a Cuntz-Pimsner algebra, which is a quotient of the Toeplitz algebra. Both algebras carry natural gauge actions of the circle, and hence one can obtain natural dynamicsbyliftingthese actions toactions of the real numbers. We study KMS states of these dynamics at, above, and below a certain critical value. For inverse temperature larger than the critical value,we finda large simplex of KMS stateson the Toeplitz algebra. For the Cuntz-Pimsner algebra, the KMS states all have inverse temperatures below the critical value. Our results for the Cuntz-Pimsner algebra overlap with recent work of Thomsen,but our proofsare quite different. At the criticalvalue, we build a KMS state of the Toeplitz algebra which factors through the Cuntz-Pimsner algebra.To understand KMS statesbelow thecritical value, we study the backward shift on the infinite path space of an ordinary directed graph.Merging our results for the Cuntz-Pimsner algebra of shifts withthe recent work about KMSstates of the graph algebras, we showthatThomsen;;s bounds on of the possible inverse temperature of KMS states are sharp. In the second component, we consider a family of *-commutinglocal homeomorphisms on a compact space and build a compactly alignedproduct system of Hilbert bimodules (in the sense of Fowler). This product system also has two interesting algebras, the Nica-Toeplitz algebra and the Cuntz-Pimsner algebra. For these algebras, the gauge action is an action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in the torus.We use the techniques from the first component of the thesisto study the KMS states for these dynamics. For large inverse temperature,we describe thesimplex of the KMS states on theNica-Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider a preferred dynamics for which there is a single critical inverse temperature, which we can normalise to be 1. We then find aKMS1 state for the Nica-Toeplitz algebra which factorsthrough the Cuntz-Pimsner algebra. We then illustrate our results by considering different backward shifts on the infinite path space of some higher-rank graphs.
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