Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = \mu(A)$ for every $t \in \Bbb{R}$ and for all borelian set A. In particular, the time-one map $\varphi^1$ also preserves the measure $\mu$ because $\mu(\varphi^{-1}(A)) = \mu(A)$.

We say that the map $\varphi^1$ is recurrent (or conservative) if all wandering set W for $\varphi^1$( meaning that $W \cap \varphi^{-n}(W) = \varnothing $ for $n \geq 1$) has necessarily $\mu(W) = 0$.

An another equivalent definition is, if A is borelian set with $\mu(A) > 0$ then $$ A \subset \displaystyle\bigcup_{n \geq 1} \varphi^{-n}(A) \ \ (mod \ \ \mu) $$

Question: Is there a complete Riemannian manifold M with infinite volume whose the time-one map of the geodesic flow is recurrent ?