Counting common perpendicular arcs in negative curvature
Parkkonen, J., & Paulin, F. (2017). Counting common perpendicular arcs in negative curvature. Ergodic Theory and Dynamical Systems, 37(3), 900-938. https://doi.org/10.1017/etds.2015.77
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Ergodic Theory and Dynamical SystemsDate
2017Copyright
© Cambridge University Press, 2016
Let D− and D+ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties
of the geodesic flow, we give an asymptotic formula as t → +∞ for the number of
common perpendiculars of length at most t from D− to D+, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of
D− and D+ of the tangent vectors at the endpoints of the common perpendiculars.
When the manifold is compact with exponential decay of correlations or arithmetic
with finite volume, we give an error term for the asymptotic. As an application, we
give an asymptotic formula for the number of connected components of the domain
of discontinuity of Kleinian groups as their diameter goes to 0.
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Cambridge University Press; London Mathematical SocietyISSN Search the Publication Forum
0143-3857Keywords
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