  In this work a quantum dynamical system $(\mathfrak M,\Phi, \varphi)$ is
constituted by a von Neumann algebra $\mathfrak M$, by a unital Schwartz map
$\Phi:\mathfrak{M\rightarrow M}$ and by a $\Phi$-invariant normal faithful
state $\varphi$ on $\mathfrak M$. The ergodic properties of a quantum dynamical
system, depends on its reversible part $(\mathfrak{D}_\infty,\Phi_\infty,
\varphi_\infty)$. It is constituted by a von Neumann sub-algebra
$\mathfrak{D}_\infty$ of $\mathfrak M$ by an automorphism $\Phi_\infty$ and a
normal state $\varphi_\infty$, the restrictions of $\Phi$ and $\varphi$ on
$\mathfrak{D}_\infty$ respectively. Moreover, if $\mathfrak{D}_\infty$ is a
trivial algebra the quantum dynamical system is ergodic. Furthermore we will
give some properties of the reversible part of quantum dynamical system, in
particular, we will study its relationships with the canonical decomposition of
Nagy-Fojas of linear contraction related to the quantum dynamical system.
