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
References:
[1]

B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal., 113 (1993), 245-317.  doi: 10.1006/jfan.1993.1052.

[2]

F. Dal'boJ.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel. J. Math., 118 (2000), 109-124.  doi: 10.1007/BF02803518.

[3]

K. Corlette and A. Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic spaces, Trans. Amer. Math. Soc., 351 (1999), 1507-1530.  doi: 10.1090/S0002-9947-99-02113-3.

[4]

M. M. DodsonM. V. MeliánD. Pestana and S. L. Velani, Patterson measure and ubiquity, Ann. Acad. Sci. Fenn. Ser. A. I Math., 20 (1995), 37-60. 

[5]

S. Edwards and H. Oh, Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds, preprint, arXiv: 2001.03377. To appear in Duke Math. J..

[6]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.

[7]

S. Galatolo, Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 2477–-2487. doi: 10.1090/S0002-9939-10-10275-5.

[8]

S. Galatolo and I. Nisoli, Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds, Nonlinearity, 24 (2011), 3099-3113.  doi: 10.1088/0951-7715/24/11/005.

[9]

A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.  doi: 10.1112/blms.12023.

[10]

S. Hersonsky and F. Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds, J. Differential Geom., 85 (2010), 271-314. 

[11]

C. T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math., 120 (1998), 691-721.  doi: 10.1353/ajm.1998.0031.

[12]

G. MargulisA. Mohammadi and H. Oh, Closed geodesics and holonomies for Kleinian manifolds, Geom. Funct. Anal., 24 (2014), 1608-1636.  doi: 10.1007/s00039-014-0299-y.

[13]

D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, Geom. Funct. Anal., 27 (2017), 1257-1287.  doi: 10.1007/s00039-017-0421-z.

[14]

D. Kelmer and S. Yu, Shrinking target problems for flows on homogeneous spaces., Trans. Amer. Math. Soc., 372 (2019), 6283-6314.  doi: 10.1090/tran/7783.

[15]

D. Kleinbock, I. Konstantoulas and F. K. Richter, Zero-one laws for eventually always hitting points in mixing systems, preprint, arXiv: 1904.08584.

[16]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[17]

D. Kleinbock and X. Zhao, An application of lattice points counting to shrinking target problems, Discrete Contin. Dyn. Syst., 38 (2018), 155-168.  doi: 10.3934/dcds.2018007.

[18]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc., 24 (2011), 603-648.  doi: 10.1090/S0894-0347-2011-00691-7.

[19]

P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Functional Analysis, 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.

[20]

J. Li and W. Pan, Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps, preprint, arXiv: 2009.12886.

[21]

F. Maucourant, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel. J. Math., 152 (2006), 143-155.  doi: 10.1007/BF02771980.

[22]

A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. (JEMS), 17 (2015), 837-897.  doi: 10.4171/JEMS/520.

[23]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), no. 95 (2005). doi: 10.24033/msmf.408.

[24]

P. Sarkar and D. Winter, Exponential mixing of frame flows for convex cocompact hyperbolic manifolds, preprint, arXiv: 2004.14551.

[25]

L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120.  doi: 10.1088/0951-7715/24/4/005.

[26]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171–202.

[27]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarthm law for geodesics, Acta Math., 149 (1982), 251-237.  doi: 10.1007/BF02392354.

[28]

B. Stratmann and S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220.  doi: 10.1112/plms/s3-71.1.197.

show all references

References:
[1]

B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal., 113 (1993), 245-317.  doi: 10.1006/jfan.1993.1052.

[2]

F. Dal'boJ.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel. J. Math., 118 (2000), 109-124.  doi: 10.1007/BF02803518.

[3]

K. Corlette and A. Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic spaces, Trans. Amer. Math. Soc., 351 (1999), 1507-1530.  doi: 10.1090/S0002-9947-99-02113-3.

[4]

M. M. DodsonM. V. MeliánD. Pestana and S. L. Velani, Patterson measure and ubiquity, Ann. Acad. Sci. Fenn. Ser. A. I Math., 20 (1995), 37-60. 

[5]

S. Edwards and H. Oh, Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds, preprint, arXiv: 2001.03377. To appear in Duke Math. J..

[6]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.

[7]

S. Galatolo, Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 2477–-2487. doi: 10.1090/S0002-9939-10-10275-5.

[8]

S. Galatolo and I. Nisoli, Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds, Nonlinearity, 24 (2011), 3099-3113.  doi: 10.1088/0951-7715/24/11/005.

[9]

A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, Bull. Lond. Math. Soc., 49 (2017), 235-245.  doi: 10.1112/blms.12023.

[10]

S. Hersonsky and F. Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds, J. Differential Geom., 85 (2010), 271-314. 

[11]

C. T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math., 120 (1998), 691-721.  doi: 10.1353/ajm.1998.0031.

[12]

G. MargulisA. Mohammadi and H. Oh, Closed geodesics and holonomies for Kleinian manifolds, Geom. Funct. Anal., 24 (2014), 1608-1636.  doi: 10.1007/s00039-014-0299-y.

[13]

D. Kelmer, Shrinking targets for discrete time flows on hyperbolic manifolds, Geom. Funct. Anal., 27 (2017), 1257-1287.  doi: 10.1007/s00039-017-0421-z.

[14]

D. Kelmer and S. Yu, Shrinking target problems for flows on homogeneous spaces., Trans. Amer. Math. Soc., 372 (2019), 6283-6314.  doi: 10.1090/tran/7783.

[15]

D. Kleinbock, I. Konstantoulas and F. K. Richter, Zero-one laws for eventually always hitting points in mixing systems, preprint, arXiv: 1904.08584.

[16]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[17]

D. Kleinbock and X. Zhao, An application of lattice points counting to shrinking target problems, Discrete Contin. Dyn. Syst., 38 (2018), 155-168.  doi: 10.3934/dcds.2018007.

[18]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc., 24 (2011), 603-648.  doi: 10.1090/S0894-0347-2011-00691-7.

[19]

P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Functional Analysis, 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.

[20]

J. Li and W. Pan, Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps, preprint, arXiv: 2009.12886.

[21]

F. Maucourant, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel. J. Math., 152 (2006), 143-155.  doi: 10.1007/BF02771980.

[22]

A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. (JEMS), 17 (2015), 837-897.  doi: 10.4171/JEMS/520.

[23]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), no. 95 (2005). doi: 10.24033/msmf.408.

[24]

P. Sarkar and D. Winter, Exponential mixing of frame flows for convex cocompact hyperbolic manifolds, preprint, arXiv: 2004.14551.

[25]

L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120.  doi: 10.1088/0951-7715/24/4/005.

[26]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171–202.

[27]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarthm law for geodesics, Acta Math., 149 (1982), 251-237.  doi: 10.1007/BF02392354.

[28]

B. Stratmann and S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220.  doi: 10.1112/plms/s3-71.1.197.

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