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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 20, 2016

On the Hausdorff Dimension of CAT(κ) Surfaces

  • David Constantine and Jean-François Lafont

Abstract

We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

References

[1] M. Bourdon. Structure conforme au bord et flot géodésique d’un CAT(-1)-espace. Enseign. Math., 41(2):63–102, 1995. Search in Google Scholar

[2] Marc Bourdon. Sur le birapport au bord des cat(-1)-espaces. Publications mathématiques de l’I.H.É.S., 83:95–104, 1996. 10.1007/BF02698645Search in Google Scholar

[3] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 1999. 10.1007/978-3-662-12494-9Search in Google Scholar

[4] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33 of Graduate Studies inMathematics. American Mathematical Society, 2001. 10.1090/gsm/033Search in Google Scholar

[5] J.W. Cannon. Shrinking cell-like decompositions of manifolds. codimension three. Annals of Mathematics, 110(1):83–112, 1979. 10.2307/1971245Search in Google Scholar

[6] Ruth M. Charney and Michael Davis. Strict hyperbolization. Topology, 34(2):329–350, 1995. 10.1016/0040-9383(94)00027-ISearch in Google Scholar

[7] Michael Davis and Tadeusz Januszkiewicz. Hyperbolization of polyhedra. J. Differential Geometry, 34(2):347–388, 1991. 10.4310/jdg/1214447212Search in Google Scholar

[8] Robert D. Edwards. Suspensions of homology spheres. Available at ArXiv:math/0610573. Search in Google Scholar

[9] Ursula Hamenstädt. Entropy-rigidity of locally symmetric spaces of negative curvature. Annals ofMathematics, 131(1):35–51, 1990. 10.2307/1971507Search in Google Scholar

[10] John G. Hocking and Gail S. Young. Topology. Dover, New York, 1988. Search in Google Scholar

[11] Anatole Katok and Boris Hasselblatt. Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995. 10.1017/CBO9780511809187Search in Google Scholar

[12] Enrico Leuzinger. Entropy of the geodesic flow for metic spaces and Bruhat-Tits buildings. Adv. Geom., 6:475–491, 2006. 10.1515/ADVGEOM.2006.029Search in Google Scholar

[13] Anthony Manning. Topological entropy for geodesic flows. Annals of Mathematics, 110(3):567–573, 1979. 10.2307/1971239Search in Google Scholar

[14] Pierre Pansu. Dimension conforme et sphère à l’infini des variétés à courbure négative. Ann. Acad. Sci. Fenn. Ser. A IMath., 14(2):177–212, 1989. 10.5186/aasfm.1989.1424Search in Google Scholar

[15] Jeremy T. Tyson and Jang-Mei Wu. Characterizations of snowflake metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math., 30(2):313–336, 2005. Search in Google Scholar

Received: 2015-3-2
Accepted: 2016-7-4
Published Online: 2016-9-20

© 2016 David Constantine and Jean-François Lafont

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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