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Hausdorff dimension of the limit set on a visibility manifold

Published online by Cambridge University Press:  17 April 2009

Hyun Jung Kim
Hyun Jung Kim
Affiliation:
Department of Mathematics, Hoseo University, Baebang Mynn, Asan 337–795, Korea, e-mail: hjkim @math.hoseo.ac.kr
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Abstract

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In this paper, for a given Fuchsian group Γ, we prove an upper estimate for the Hausdorff dimension of the radial limit set in the visibility manifold. Further, if Γ is a convex cocompact group, we find the exact Hausdroff dimension of the limit set.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 199 - 209
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, Progress in Mathematics 61 (Birkhauser Boston Inc, Boston MA, 1985).CrossRefGoogle Scholar
[2]Eberlein, P., ‘Geodesic flow in certain manifolds without conjugate points’, Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
[3]Eberlein, P. and O'Neil, B., ‘Visibility manifolds’, Pacific J. Math. 46 (1973), 45–109.CrossRefGoogle Scholar
[4]Hamenstädt, U., ‘A new description of the Bowen-Margulis measure’, Ergodic Theory Dynamical Systems 9 (1989), 455–464.CrossRefGoogle Scholar
[5]Kaimanovich, V.A., ‘Invariant measure for the geodesic flow and measures at infinity on negatively curved manifolds’, Ann Inst. H. Poincaré Phys. Théor. 53 (1990), 361–393.Google Scholar
[6]Kim, V.H., ‘Conformal density of visibility manifold’, Bull. Korean Math. Soc. 38 (2001), 211–222.Google Scholar
[7]Patterson, S.J., Lectures on measures on limit sets on Kleinian group, London Math. Soc. Lecute Notes Ser. 111 (Cambridge University Press, Cambridge, 1987), pp. 281–323.Google Scholar
[8]Sullivan, D., ‘The density at infinity of a discrete group of hyperbolic motions’, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202.CrossRefGoogle Scholar
[9]Yue, C., ‘The ergodic theory of discrete isometry groups on manifolds of variable negative curvature’, Trans. Amer. Math. Soc. 348 (1996), 4965–5005.CrossRefGoogle Scholar