Assume we have a smooth compact manifold $M$ with boundary and a smooth complete vector field $X$ on $M$. Let $\phi^{t}$ be the resulting flow and let $\mu$ be a probability measure on $M$. Define the Poincare section by $\Psi: \partial M \to \partial M$, where $\partial M$ denotes the boundary of $M$ and let $\mu_{\partial M}$ be the conditioned measure. Assume the cocycle $(\Psi,\mu_{\partial M}, D\Psi)$ to be ergodic.
Under which conditions does that imply ergodicity for the flow cocycle $(\phi^{t},\mu,D\phi^{t})$?
I am especially interested in systems, which are already nonuniformly hyperbolic.
Thanks in advance!