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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


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.


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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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