On the ergodic decomposition for a cocycle

International audienceLet $(X, {\goth X}, \mu, \tau)$ be an ergodic dynamical system and $\varphi$ be a measurable map from $X$ to a locally compact second countable group $G$ with left Haar measure $m_G$. We consider the map $\tau_\varphi$ defined on $X \times G$ by $\tau_\varphi: (x,g) \rightarrow (\tau x, \varphi(x)g)$ and the cocycle $(\varphi_n)_{n \in \mathbb{Z}}$ generated by $\varphi$. Using a characterization of the ergodic invariant measures for $\tau_\varphi$ (\cite{Ra06}), we give the form of the ergodic decomposition of $\mu(dx) \otimes m_G(dg)$ or more generally of the $\tau_\varphi$-invariant measures $\mu_\chi(dx) \otimes \chi(g) m_G(dg)$, where $\mu_\chi(dx)$ is $\chi\circ \varphi$-conformal for an exponential $\chi$ on $G$