Nihonkai Mathematical Journal

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

Nobuhiro Innami

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 $\pi(G_{t_n}(x_n)) \rightarrow q$.

Article information

Source
Nihonkai Math. J., Volume 23, Number 1 (2012), 43-55.

Dates
First available in Project Euclid: 5 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1352124687

Mathematical Reviews number (MathSciNet)
MR3014413

Zentralblatt MATH identifier
1269.53038

Subjects
Primary: 53C22
Secondary: 58E10: Applications to the theory of geodesics (problems in one independent variable)

Citation

Innami, Nobuhiro. The asymptotic behavior of geodesic crcles in 2-torus of revolution and a sub-ergodic property. Nihonkai Math. J. 23 (2012), no. 1, 43--55. https://projecteuclid.org/euclid.nihmj/1352124687