Let , , be a Markov chain on (, with Polish state space E, and let . Let H be some function defined on the set of probability measures on E with values in [- which is nice enough. Transformed laws are defined by
The possible limit laws of under are described. The main assumption is that is uniformly ergodic. Roughly speaking, the limit laws are mixtures of Markov chains minimizing a certain free energy. The method of proof strongly relies on large deviation techniques.