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Invariant curves of smooth quasi-periodic mappings
An application of lattice points counting to shrinking target problems
Brandeis University, Waltham MA, 02454, USA |
We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.
References:
| [1] |
J. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. (Math. Sci.), 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
| [2] |
_____, Cusp excursions on parameter spaces, J. Lond. Math. Soc. , 87 (2013), 741–765 |
| [3] |
Y. Benoist and H. Oh,
Effective equidistribution of $S$-integral points on symmetric varieties, Annales de L'Institut Fourier, 62 (2012), 1889-1942.
doi: 10.5802/aif.2738. |
| [4] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/BF02809888. |
| [5] |
D. Dolgopyat,
Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
| [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] |
A. Gorodnik and A. Nevo,
Counting lattice points, J. Reine Angew. Math., 663 (2012), 127-176.
doi: 10.1515/CRELLE.2011.096. |
| [8] |
A. Gorodnik and N. Shah,
Khinchin's theorem for approximation by integral points on quadratic varieties, Math. Ann., 350 (2011), 357-380.
doi: 10.1007/s00208-010-0561-z. |
| [9] |
N. Haydn, M. Nicol, T. Persson and S. Vaienti,
A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.
doi: 10.1017/S014338571100099X. |
| [10] |
H. Huber,
Über eine neue Klasse automorpher Functionen und eine Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv., 30 (1956), 20-62.
doi: 10.1007/BF02564331. |
| [11] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
| [12] |
P. Lax and R. Phillips,
The asymptotic distribution of lattice points in Euclidean and Non-Euclidean spaces, J. Funct. Anal., 46 (1982), 280-350.
doi: 10.1016/0022-1236(82)90050-7. |
| [13] |
F. Maucourant,
Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.
doi: 10.1007/BF02771980. |
| [14] |
C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in: Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, CA, 1984), 163–181, Math. Sci. Res. Inst. Publ. 6, Springer, New York, 1987.
doi: 10.1007/978-1-4612-4722-7_6. |
| [15] |
W. Philipp,
Some metrical theorems in number theory, Pacific J. Math., 20 (1967), 109-127.
doi: 10.2140/pjm.1967.20.109. |
| [16] |
M. Ratner,
The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.
doi: 10.1017/S0143385700004004. |
| [17] |
D. Sullivan,
Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
show all references
References:
| [1] |
J. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. (Math. Sci.), 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
| [2] |
_____, Cusp excursions on parameter spaces, J. Lond. Math. Soc. , 87 (2013), 741–765 |
| [3] |
Y. Benoist and H. Oh,
Effective equidistribution of $S$-integral points on symmetric varieties, Annales de L'Institut Fourier, 62 (2012), 1889-1942.
doi: 10.5802/aif.2738. |
| [4] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/BF02809888. |
| [5] |
D. Dolgopyat,
Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
| [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] |
A. Gorodnik and A. Nevo,
Counting lattice points, J. Reine Angew. Math., 663 (2012), 127-176.
doi: 10.1515/CRELLE.2011.096. |
| [8] |
A. Gorodnik and N. Shah,
Khinchin's theorem for approximation by integral points on quadratic varieties, Math. Ann., 350 (2011), 357-380.
doi: 10.1007/s00208-010-0561-z. |
| [9] |
N. Haydn, M. Nicol, T. Persson and S. Vaienti,
A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.
doi: 10.1017/S014338571100099X. |
| [10] |
H. Huber,
Über eine neue Klasse automorpher Functionen und eine Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv., 30 (1956), 20-62.
doi: 10.1007/BF02564331. |
| [11] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
| [12] |
P. Lax and R. Phillips,
The asymptotic distribution of lattice points in Euclidean and Non-Euclidean spaces, J. Funct. Anal., 46 (1982), 280-350.
doi: 10.1016/0022-1236(82)90050-7. |
| [13] |
F. Maucourant,
Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.
doi: 10.1007/BF02771980. |
| [14] |
C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in: Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, CA, 1984), 163–181, Math. Sci. Res. Inst. Publ. 6, Springer, New York, 1987.
doi: 10.1007/978-1-4612-4722-7_6. |
| [15] |
W. Philipp,
Some metrical theorems in number theory, Pacific J. Math., 20 (1967), 109-127.
doi: 10.2140/pjm.1967.20.109. |
| [16] |
M. Ratner,
The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.
doi: 10.1017/S0143385700004004. |
| [17] |
D. Sullivan,
Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
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