Geodesic
A note on the shrinking sector problem for surfaces of variable negative curvature
Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 281-291 
Mark Pollicott. A note on the shrinking sector problem for surfaces of variable negative curvature. Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 281-291. http://geodesic.mathdoc.fr/item/TRSPY_2017_297_a14/
@article{TRSPY_2017_297_a14,
     author = {Mark Pollicott},
     title = {A note on the shrinking sector problem for surfaces of variable negative curvature},
     journal = {Informatics and Automation},
     pages = {281--291},
     year = {2017},
     volume = {297},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_297_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Given the universal cover $\widetilde V$ for a compact surface $V$ of variable negative curvature and a point $\widetilde x_0\in\widetilde V$, we consider the set of directions $\widetilde v\in S_{\widetilde x_0}\widetilde V$ for which a narrow sector in the direction $\widetilde v$, and chosen to have unit area, contains exactly $k$ points from the orbit of the covering group. We can consider the size of the set of such $\widetilde v$ in terms of the induced measure on $S_{\widetilde x_0}\widetilde V$ by any Gibbs measure for the geodesic flow. We show that for each $k$ the size of such sets converges as the sector grows narrower and describe these limiting values. The proof involves recasting a similar result by Marklof and Vinogradov, for the particular case of surfaces of constant curvature and the volume measure, by using the strong mixing property for the geodesic flow, relative to the Gibbs measure.

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