# American Institute of Mathematical Sciences

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

## Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

 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

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

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.

Citation: Dubi Kelmer, Hee Oh. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds. Journal of Modern Dynamics, 2021, 17: 401-434. doi: 10.3934/jmd.2021014
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##### References:
 [1] Brent Everitt, John Ratcliffe and Steven Tschantz. The smallest hyperbolic 6-manifolds. Electronic Research Announcements, 2005, 11: 40-46. [2] Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553 [3] Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018 [4] Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008 [5] Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451 [6] Walter D. Neumann and Jun Yang. Invariants from triangulations of hyperbolic 3-manifolds. Electronic Research Announcements, 1995, 1: 72-79. [7] Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009 [8] Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133 [9] Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Tori with hyperbolic dynamics in 3-manifolds. Journal of Modern Dynamics, 2011, 5 (1) : 185-202. doi: 10.3934/jmd.2011.5.185 [10] Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006 [11] Jörg Schmeling. A notion of independence via moving targets. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 269-280. doi: 10.3934/dcds.2006.15.269 [12] Dmitri Burago, Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups. Journal of Modern Dynamics, 2008, 2 (4) : 541-580. doi: 10.3934/jmd.2008.2.541 [13] Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641 [14] Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013 [15] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [16] Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307 [17] Thierry Daudé, Damien Gobin, François Nicoleau. Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds. Inverse Problems & Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016 [18] Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089 [19] Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111 [20] Sara D. Cardell, Amparo Fúster-Sabater. Modelling the shrinking generator in terms of linear CA. Advances in Mathematics of Communications, 2016, 10 (4) : 797-809. doi: 10.3934/amc.2016041

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