Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T04:09:31.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounded geodesics in rank-1 locally symmetric spaces

Published online by Cambridge University Press:  14 October 2010

C. S. Aravinda  and
Enrico Leuzinger
C. S. Aravinda
Affiliation:
School of Mathematics, TIFR, Homi Bhabha Road, Bombay-400005, India
Enrico Leuzinger
Affiliation:
Mathematisches Institut, Universitat Zurich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland
Get access Rights & Permissions [Opens in a new window]

Abstract

Let M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p ∈ M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension 1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 5 , October 1995 , pp. 813 - 820
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A] Aravinda, C. S.. Bounded geodesies and Hausdorff dimension. Math. Proc. Cam. Phil. Soc. To appear.Google Scholar
[B] Brehm, U.. The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom. Dedicata 33 (1990), 59–76.CrossRefGoogle Scholar
[D] Dani, S. G.. Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61 (1986), 636–660.CrossRefGoogle Scholar
[E] Eberlein, P.. Lattices in spaces of nonpositive curvature. Ann. Math. III (1980), 435–476.CrossRefGoogle Scholar
[HI] Heintze, E. and Hof, H. C. Im. Geometry of horospheres. J. Diff. Geom. 12 (1977), 481–491.Google Scholar
[H] Hsiang, W. Y.. On the trigonometry of two-point homogeneous spaces. Ann. Global Anal. Geom. 7 (1989), 25–45.Google Scholar
[L] Leuzinger, E.. Les lois des sinus des espaces symetriques S de rang 1 et 1'application moment de T.S. C. R. Acad. Sci. Paris 312 (1991), 605–608.Google Scholar
[Ma] Mautner, F. I.. Geodesic flows on symmetric Riemann spaces. Ann. Math. 65 (1957), 416–431.CrossRefGoogle Scholar
[Mo] Moore, C. C.. Ergodicity of flows on homogeneous spaces. Amer. J. Math. 88 (1966), 154–178.[S]CrossRefGoogle Scholar
Schmidt, W. M.. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178–199.CrossRefGoogle Scholar