The mini-course covers:
Breiman Ergodic Theorem, which states that for any Markov chain on a compact metric space with a continuous transition probability, the empirical measures accumulate onto stationary measures. Its proof will use transfer operator (acting on L^1 observables), Markov operator (acting on probability measures), and a martingale argument due to Furstenberg.
Invariant measures for skew-products with one-sided shift in the base vs. with two-sided shift in the base. Its proof will use “disintegration” of a measure, and another martingale argument.
U-state measures, and their correspondence with stationary measures.
Potentially, more results on “synchronization”, from the theoretical Random Dynamical Systems point of view.
This mini-course shall be very accessible to anyone who is proficient in Real Analysis (and some basic Measure Theory). In particular, you do not need to know everything or anything listed above; they are there just in case some key words might motivate you more than the mere title. Also, the Probability Theory tools will be supplied as needed.