We study the absolute continuity of ergodic measures of Markov chains $X_{n+1}=F(X_n,Y_{n+1})$ for the discrete case, and $dX_t=b(X_t)dt+\sigma(X_t).dW_t$ for the continuous case. In the discrete case, we provide with a method enabling to deal with the case where the chains has several invariant measures whereas previous works (c.f. \cite{coquio1992calcul,gravereaux1988calcul}) made assumptions of contractivity, and hence unique ergodicity. Besides, the smoothness assumptions on $F$ are weakened. In the continuous case, we make stronger smoothness assumptions than \cite{bogachev2009elliptic}, but non-degeneracy assumptions are strongly weakened. The proofs are based on Dirichlet forms theory, and ergodic theory arguments.
