Piraino, Mark2019-07-162019-07-1620192019-07-16http://hdl.handle.net/1828/10974We study two problems. The first concerning ergodic properties of measures on $\S^{\Z}$ such that $\mu_{\A,t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\norm{A_{x_{0}}\cdots A_{x_{n-1}}}^{t}$ where $\A=(A_{0},\ldots, A_{M-1})$ is a collection of matrices, such measures are known as matrix Gibbs states. In particular we give a sufficient condition for $\mu_{\A,t}$ to be isomorphic to a Bernoulli shift and mix at an exponential rate. The second problem concerns factors of Gibbs states. In particular we show that all of classical uniqueness regimes for Gibbs states are closed under factor maps which satisfy a mixing in fibers condition. The unifying approach to both of these problems is to realize the measure of cylinder sets in terms of positive operators.enAvailable to the World Wide Webmatrix Gibbs stateshidden Markov measuresthermodynamic formalismThesis