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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 ?

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Take a compact, connected Riemannian manifold $M$ with negative sectional curvature. Then choose any cover $M'$ of $M$ which is connected, Galois, and whose group of deck transformations is $\mathbb{Z}$ or $\mathbb{Z}^2$.

In dimension $2$ and for a $\mathbb{Z}$ cover, this can be done by taking $\mathbb{Z}$ copies of the manifold $M$, cutting along an essential curve (in dimension $2$). Then we get manifold with boundaries which can be indexed by $\mathbb{Z} \times \{0,1\}$. Glue $(p,0)$ with $(p+1,1)$.

The geodesic flow on $T^1 M'$ behaves roughly like a random walk on $\mathbb{Z}$ (or $\mathbb{Z}^2$) with centered and bounded increments, so it is recurrent.

There are other examples in the same fashion, for instance $\mathbb{C} \setminus \mathbb{Z}$ endowed with a $\mathbb{Z}$-invariant hyperbolic metric.

See for instance:

  • J. Aaronson, M. Denker, Distributional limits for hyperbolic, infinite volume geodesic flows (Tr. Mat. Inst. Steklova 216 (1997), Din. Sist. i Smezhnye Vopr., 181--192)

  • J. Aaronson, M. Denker, The Poincar\'e series of $\mathbb{C}\setminus\mathbb{Z}$ (Ergodic Theory Dynam. Systems 19 (1999), no. 1, 1--20.)

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