Share This Article:

Volume of Geodesic Balls in Finsler Manifolds of Hyperbolic Type

Abstract Full-Text HTML Download Download as PDF (Size:2579KB) PP. 391-399
DOI: 10.4236/apm.2014.48050    5,000 Downloads   5,379 Views  

ABSTRACT

Let  be a compact Finsler manifold of hyperbolic type, and  be its universal Finslerian covering. In this paper we show that the growth function of the volume of geodesic balls of  is of purely exponential type.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ogouyandjou, C. (2014) Volume of Geodesic Balls in Finsler Manifolds of Hyperbolic Type. Advances in Pure Mathematics, 4, 391-399. doi: 10.4236/apm.2014.48050.

References

[1] Margulis, M.A. (1969) Applications of Ergotic Theory to the Investigation of Manifolds of Negative Curvature. Functional Analysis and Its Applications, 3, 335-336. http://dx.doi.org/10.1007/BF01076325
[2] Manning, A. (1979) Topological Entropy for Geodesic Flows. Annals of Mathematics, 110, 567-573.
http://dx.doi.org/10.2307/1971239
[3] Knieper, G. (2002) Hyperbolic Dynamics and Riemannian Geometry. In: Hasselblatt, B. and Katok, A., Eds., Handbook of Dynamical Systems 1A, Elsevier Science, Amsterdam, 453-545.
[4] Ezin, J.-P. and Ogouyandjou, C. (2005) The Growth Function of the Volume of Geodesic Balls in Riemannian Manifolds of Hyperbolic Type. IMHOTEP, 6, 9-17.
[5] Bao, D., Chern, S.-S. and Shen, Z. (2000) An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics 200. Communication in Contemporay Math, No. 4, Springer-Verlag, 511-533.
[6] Shen, Z. (1977) Volume Comparison and Its Applications in Riemann-Finsler Geometry. Advances in Mathematics, 128, 306-328. http://dx.doi.org/10.1006/aima.1997.1630
[7] Zaustinsky, E.M. (1962) Extremals on Compact E-Surfaces. Transactions of the American Mathematical Society, 102, 433-445.
[8] Klingenberg, W. (1971) Geodatischer Fluss auf Mannigfaltigkeiten vom hyperbolishen Typ. Inventiones Mathematicae, 14, 63-82. http://dx.doi.org/10.1007/BF01418743
[9] Schroder, J.P. (2014) Minimal Rays on Surfaces of Genus Greater than One. arXiv:1404.0573v1 [math.DS]
[10] Ancona, A. (1988) Théorie du potentiel sur les graphes et les varieties. In: Ancona, A., et al., Eds., Potential Theory, Surveys and Problems, Lecture Notes in Mathematics, No. 1344, Springer-Verlag, Berlin.
[11] Coornaert, M., Delzant, T. and Papadopoulos, A. (1990) Géométrie et théorie des groups. Lecture Notes in Mathematics. No. 1441, Springer-Verlag, Berlin.
[12] Coornert, M. and Papadoupoulos, A. (1993) Symbolic Dynamics and Hyperbolic Groups. Lecture Notes in Mathematics, No. 1539, Springer-Verlag, Berlin.
[13] Coornaert, M. (1993) Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific Journal of Mathematics, 159, 241-270. http://dx.doi.org/10.2140/pjm.1993.159.241
[14] Cao, J. (2000) Cheeger Isoperimetric Constants of Gromov Hyperbolic Spaces with Quasi-Pole. Communications in Contemporary Mathematics, 2, 511-533.
http://dx.doi.org/10.1142/S0219199700000232
[15] Patterson, S. (1976) The Limit Set of Fuchsian Group. Acta Mathematica, 163, 241-273.
http://dx.doi.org/10.1007/BF02392046

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.