A Note on the Shrinking Sector Problem for Surfaces of Variable Negative Curvature

作者：Pollicott Mark^*

来源：Proceedings of the Steklov Institute of Mathematics, 2017, 297(1): 254-263.

DOI：10.1134/S0081543817040150

Given the universal cover (V) over tilde V for a compact surface V of variable negative curvature and a point (x) over tilde (0) is an element of (V) over tilde , we consider the set of directions (v) over tilde is an element of S-(x) over tilde0 (v) over tilde for which a narrow sector in the direction (v) over tilde, 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 (v) over tilde in terms of the induced measure on S-(x) over tilde0 (V) over tilde 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.

• 出版日期2017-5