  We study the following model of hidden Markov chain: $Y_i=X_i+\epsilon_i$, $
i=1,...,n+1$ with $(X_i)$ a real-valued positive recurrent and stationary
Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise independent of the
sequence $(X_i)$ having a known distribution. We present an adaptive estimator
of the transition density based on the quotient of a deconvolution estimator of
the density of $X_i$ and an estimator of the density of $(X_i,X_{i+1})$. These
estimators are obtained by contrast minimization and model selection. We
evaluate the $L2$ risk and its rate of convergence for ordinary smooth and
supersmooth noise with regard to ordinary smooth and supersmooth chains. Some
examples are also detailed.
