This thesis consists of two independent chapters. Each chapter has its own detailed introduction and references. In Chapter one we give new complete topological conjugacy invariants for finite state stationary Markov chains. These new invariants give a classification up to a topological conjugacy which preserves certain equilibrium states. We use our new invariants to investigate Williams' problem. Finally, we generalise the topological classification of onesided finite state stationary Markov chains to give a classification up to block-isomorphism. In Chapter two we investigate a problem posed by Shub and Sullivan on the classification of real analytic Lebesgue measure-preserving expanding endomorphisms of the circle. We introduce a new Jacobian invariant that enables us to study the phase factor. Finally, we introduce complete isomorphism invariants but these invariants have a measure-theoretic and topological nature.