Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Dubi Kelmer; Hee Oh

doi:10.3934/jmd.2021014

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2021, Volume 17: 401-434. Doi: 10.3934/jmd.2021014

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Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Dubi Kelmer^1, and
Hee Oh^2,

1.
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA
2.
Department of Mathematics, Yale University, New Haven, CT 06511, USA

Received: January 25, 2020

Revised: March 15, 2021

Published: October 2021

DK: Partially supported by NSF CAREER grant DMS-1651563.
HO: Partially supported by NSF grants.

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Abstract

Let $ \mathscr{M} $ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.

Keywords:

Shrinking targets,
hyperbolic manifolds.

Mathematics Subject Classification: Primary: 37A17; Secondary: 20F67, 22E40.

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References

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