On the ergodic decomposition for a cocycle
Volume 117 / 2009
Colloquium Mathematicum 117 (2009), 121-156
MSC: 28D05, 37A20, 37A40, 37B20.
DOI: 10.4064/cm117-1-8
Abstract
Let $(X, {\mathfrak X}, \mu, \tau)$ 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 $\tau_\varphi: (x,g) \mapsto (\tau x, \varphi(x)g)$ and the cocycle $(\varphi_n)_{n \in \mathbb{Z}}$ generated by $\varphi$. Using a characterization of the ergodic invariant measures for $\tau_\varphi$, 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 $G$.
