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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
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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

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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