Connection between Ergodic Theory and Markov Chains

Question

Could someone suggest a good reference where the connection between Ergodic Theory and (ergodic) Markov Chains is nicely explained ?

Answer 1

The best reference is of course the Katok-Hasselblatt Cambridge University Press book, although it really depends on what you already know: their book is useless if you are beginning to study the theory. If you want a less intimidating reference, you will profit from the much more down to earth Brin-Stuck version (for the most part the proofs are really the same in the two books, although Brin is a master of ergodicity and of course you will get an account on google's algorithm; incidentally, one of the founders of Google is Brin's son).

However, you don't tells us if you are talking about topological Markov chains or about Markov measures. As a departure point for something more interesting, one could observe that the measure induced on the symbolic dynamics by the Lebesgue measure and a linear hyperbolic toral automorphism via (any) Markov partition is a Markov measure, which may be what you are looking for, or perhaps something related.