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An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms
Logarithm laws for unipotent flows on hyperbolic manifolds
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA |
We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.
References:
| [1] |
L. Ahlfors,
On the fixed points of Möbius tranformations in $\mathbb{R}^{n}$, Annales Academiae, 10 (1985), 15-27.
doi: 10.5186/aasfm.1985.1005. |
| [2] |
J. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
| [3] |
J. Athreya,
Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765.
doi: 10.1112/jlms/jds074. |
| [4] |
J. Athreya and G. Margulis,
Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
| [5] |
J. Athreya and G. Margulis,
Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16.
doi: 10.3934/jmd.2017001. |
| [6] |
A. Borel,
Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560.
doi: 10.4310/jdg/1214430642. |
| [7] |
J. Elstrodt, F. Grunewald and J. Mennicke,
Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390.
doi: 10.1007/BF01200359. |
| [8] |
J. Elstrodt, F. Grunewald and J. Mennicke,
Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent. Math., 101 (1990), 641-685.
doi: 10.1007/BF01231519. |
| [9] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326.
doi: 10.2307/1970838. |
| [10] |
P. Garrett, Harmonic analysis on spheres, http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/09_spheres.pdf. |
| [11] |
V. Gritsenko,
Arithmetic of quaternions and Eisenstein Series, translation in J. Soviet Math., 52 (1990), 3056-3063.
doi: 10.1007/BF02342923. |
| [12] |
D. Kelmer and A. Mohammadi,
Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784.
doi: 10.1007/s00039-012-0181-8. |
| [13] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
| [14] |
A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. |
| [15] |
R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., SLN 544, Berlin-Heidelberg-New York, 1976. |
| [16] |
J. R. Parker, Hyperbolic spaces, Jyväskylä Lectures in Mathematics 2, 2008. |
| [17] |
C. D. Sogge,
Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
| [18] |
D. Sullivan,
Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
| [19] |
T. Ton-That,
Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46.
doi: 10.1090/S0002-9947-1976-0399366-1. |
| [20] |
G. Warner, Selberg's trace formula for non-uniform lattices: The $\mathbb{R}$-rank one case, in Studies in Algebra and Number Theory, Adv. in Math. Suppl. Stud., 6, Academic Press, New York-London, 1979, 1-142. |
show all references
References:
| [1] |
L. Ahlfors,
On the fixed points of Möbius tranformations in $\mathbb{R}^{n}$, Annales Academiae, 10 (1985), 15-27.
doi: 10.5186/aasfm.1985.1005. |
| [2] |
J. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
| [3] |
J. Athreya,
Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765.
doi: 10.1112/jlms/jds074. |
| [4] |
J. Athreya and G. Margulis,
Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
| [5] |
J. Athreya and G. Margulis,
Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16.
doi: 10.3934/jmd.2017001. |
| [6] |
A. Borel,
Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560.
doi: 10.4310/jdg/1214430642. |
| [7] |
J. Elstrodt, F. Grunewald and J. Mennicke,
Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390.
doi: 10.1007/BF01200359. |
| [8] |
J. Elstrodt, F. Grunewald and J. Mennicke,
Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent. Math., 101 (1990), 641-685.
doi: 10.1007/BF01231519. |
| [9] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326.
doi: 10.2307/1970838. |
| [10] |
P. Garrett, Harmonic analysis on spheres, http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/09_spheres.pdf. |
| [11] |
V. Gritsenko,
Arithmetic of quaternions and Eisenstein Series, translation in J. Soviet Math., 52 (1990), 3056-3063.
doi: 10.1007/BF02342923. |
| [12] |
D. Kelmer and A. Mohammadi,
Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784.
doi: 10.1007/s00039-012-0181-8. |
| [13] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
| [14] |
A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. |
| [15] |
R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., SLN 544, Berlin-Heidelberg-New York, 1976. |
| [16] |
J. R. Parker, Hyperbolic spaces, Jyväskylä Lectures in Mathematics 2, 2008. |
| [17] |
C. D. Sogge,
Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
| [18] |
D. Sullivan,
Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
| [19] |
T. Ton-That,
Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46.
doi: 10.1090/S0002-9947-1976-0399366-1. |
| [20] |
G. Warner, Selberg's trace formula for non-uniform lattices: The $\mathbb{R}$-rank one case, in Studies in Algebra and Number Theory, Adv. in Math. Suppl. Stud., 6, Academic Press, New York-London, 1979, 1-142. |
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