Some Probabilistic Properties of General Topological Markov Chains

In this paper, we utilize the framework of Markov processes to attain a more probabilistic perspective on the theory of transfer operators. In doing so, we establish a functional central limit theorem (FLCT) for an $O(N)$ model associated with Dyson potential on the one-dimensional lattice. We also proof a FLCT on a non-compact alphabet setting, for a model associated with a Dyson type potential on the one-dimensional lattice. A Breimann ergodic theorem for equilibrium measures arising from the Ruelle Operator Formalism is proved. Furthermore, we obtain a qualitative criterion (strong transitivity) to determine when the conformal measure has full support. As an application, we show how to connect the Ruelle operator framework with the perspective of the Hopf theory of Markov Processes.