Cover & Thomas, 2nd Edition, Jul 2006, pg 81, section 4.4 - entropy rate of markov processes, I did not remember all the conditions needed for this to hold, please read if you are interesting.

[0] staff.ustc.edu.cn/~cgong821 /Wiley.Interscience.Elements.of.Information.Theory.Jul.2006.eBook-DDU.pdf seems to have a copy indexed by Google. I suspect it is not legitimate