  In this paper a new estimator for the transition density $\pi$ of an
homogeneous Markov chain is considered. We introduce an original contrast
derived from regression framework and we use a model selection method to
estimate $\pi$ under mild conditions. The resulting estimate is adaptive with
an optimal rate of convergence over a large range of anisotropic Besov spaces
$B_{2,\infty}^{(\alpha_1,\alpha_2)}$. Some simulations are also presented.
