The asymptotic behavior of geodesic crcles in 2-torus of revolution and a sub-ergodic property

Abstract

Let $M$ be a complete Riemannian manifold with finite volume and $G_t$ the geodesic flow on the unit tangent bundle $SM$. In the light of the Poincaré recurrence property we study the following properties. (P1) For any point $p \in M$ and any open set $ U \subset M $ there exists an $R > 0$ such that $\pi(G_t(S_pM)) \cap U \neq \emptyset$ for all $t > R$. (P2) For any unit tangent vector $x \in SM$ and any point $q \in M$ there exist a sequence of unit tangent vectors $x_n \in SM$ and a sequence $t_n \rightarrow \infty$ such that $x_n \rightarrow x$ and $\pi(G_{t_n}(x_n)) \rightarrow q$.