A note on the shrinking sector problem for surfaces of variable negative curvature

Abstract

Given the universal cover Ṽ for a compact surface V of variable negative curvature and a point x̃₀ ∈ Ṽ, we consider the set of directions \({\widetilde v_0} \in {S_{\widetilde {{x_0}}}}\widetilde V\) for which a narrow sector in the direction ṽ, 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 ṽ 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.