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2012 The asymptotic behavior of geodesic crcles in 2-torus of revolution and a sub-ergodic property
Nobuhiro Innami
Nihonkai Math. J. 23(1): 43-55 (2012).

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

Citation

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Nobuhiro Innami. "The asymptotic behavior of geodesic crcles in 2-torus of revolution and a sub-ergodic property." Nihonkai Math. J. 23 (1) 43 - 55, 2012.

Information

Published: 2012
First available in Project Euclid: 5 November 2012

zbMATH: 1269.53038
MathSciNet: MR3014413

Subjects:
Primary: 53C22
Secondary: 58E10

Keywords: Billiard , circle , geodesic flow , pole , Torus

Rights: Copyright © 2012 Niigata University, Department of Mathematics

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Vol.23 • No. 1 • 2012
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