In this paper, we study first the problem of nonparametric estimation of the stationary density of a discrete-time Markov chain . We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density of and so to provide an adaptive estimator of the transition density . We give bounds in norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.