Abstract

In this paper we consider two counting problems associated with compact negatively curved surfaces and improve classical asymptotic estimates due to Margulis. In the first, we show that the number of closed geodesics of length at most T has an exponential error term. In the second we show that the number of geodesic arcs (between two fixed points x and y) of length at most T has an exponential error term. The proof is based on a detailed study of the zeta function and Poincaré series and benefits from recent work of Dolgopiat.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1019-1042
Launched on MUSE
1998-10-01
Open Access
No
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