irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant distribution over the states. (In the Markov chain case, each of the ergodic components corresponds to an irreducible sub-space.)
By "ergodic processes", I understand it to be the same as "ergodic measure-preserving dynamic system", if I am correct.
As far as I know an ergodic measure-preserving dynamic system is a mapping $\Phi: T \times S \to S$ that satisfies a couple of properties, where $S$ is the state space, and $T$ is the time space. Sometimes there is a measure preserving mapping on $S$ that can generate the system by repeating itself.
So I wonder how a Markov chain can be written as a mapping $\Phi: T \times S \to S$, and what the measure preserving mapping that generates the Markov chain is?