In my previous post asked here http://stats.stackexchange.com/questions/173024/introduction-to-markov-process-how-to-prove-that-a-process-is-markov-part-1, -- a process is Markovian if it follows the memoryless property. Consider, a dynamical system $x_n = f(x_{n-1})$ and a (stationary) Markov chain ${(X_n)}_{n \in \mathbb{Z}}$ in discrete time with each $X_n$ taking its values in a finite set $E$. The *canonical space* of the Markov chain is the product set $E^{\mathbb{Z}}$. The trajectory $X=(\ldots, X_{-1}, X_{0}, X_1, \ldots)$ of the Markov chain is a random variable taking its values in $E^{\mathbb{Z}}$. Let $\mu$ be its invariant distribution.


Question1: How does one show that a Markov Process generates (iid) random variables or is it only random variables which may not be iid?

Question2: A simplified version of explanation as to how a Markov Chain is a dynamical system and vice-versa (how a dynamical system can be a Markov Chain).

Links to Theorum and proof will be additionally very helpful