On the ergodic decomposition for a cocycle

Jean-Pierre Conze ¹   ; Albert Raugi ²

¹ IRMAR, CNRS UMR 6625 Université de Rennes I Campus de Beaulieu 35042 Rennes Cedex, France
² Université de Rennes I Campus de Beaulieu 35042 Rennes Cedex, France

Colloquium Mathematicum, Tome 117 (2009) no. 1, pp. 121-156

Résumé

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 $G$.

Détail

DOI : 10.4064/cm117-1-8

Keywords: mathfrak tau ergodic dynamical system varphi measurable map locally compact second countable group haar measure consider map tau varphi defined times tau varphi mapsto tau varphi cocycle varphi mathbb generated nbsp varphi using characterization ergodic invariant measures tau varphi form ergodic decomposition otimes generally tau varphi invariant measures chi otimes chi where chi chi circ varphi conformal exponential chi nbsp