2016, 10: 1-21. doi: 10.3934/jmd.2016.10.1

Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$

1. 

Department of Mathematics, The Ohio State University, 231 W. 18th Ave., MA 350, Columbus, OH 43210, United States

Received  May 2015 Revised  October 2015 Published  February 2016

In this note, we consider the orbits $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $\{u(t)\}$ is the standard unipotent one-parameter subgroup in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on~the initial point $p$, we can prove that the trajectory $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh [22].
Citation: Cheng Zheng. Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$. Journal of Modern Dynamics, 2016, 10: 1-21. doi: 10.3934/jmd.2016.10.1
References:
[1]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.

[2]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.

[3]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for fuchsian groups, Duke Math. J., 51 (1984), 185-194. doi: 10.1215/S0012-7094-84-05110-X.

[4]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[5]

H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 95-115.

[6]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbb R$-rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.

[8]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997.

[9]

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

[10]

G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, 377-398.

[11]

M. V. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93. doi: 10.1307/mmj/1029004675.

[12]

M. Ratner, The rate of mixing for geodesic and horocycle flow, Ergodic Theory Dyn. Syst., 7 (1987), 267-288. doi: 10.1017/S0143385700004004.

[13]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.

[14]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.

[15]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357.

[16]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.

[17]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618. doi: 10.1016/j.matpur.2014.07.004.

[18]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.

[19]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732. doi: 10.1215/S0012-7094-94-07521-2.

[20]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328. doi: 10.3934/jmd.2013.7.291.

[21]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, arXiv:1501.05228.

[22]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.

show all references

References:
[1]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.

[2]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.

[3]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for fuchsian groups, Duke Math. J., 51 (1984), 185-194. doi: 10.1215/S0012-7094-84-05110-X.

[4]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[5]

H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 95-115.

[6]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbb R$-rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.

[8]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997.

[9]

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

[10]

G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, 377-398.

[11]

M. V. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93. doi: 10.1307/mmj/1029004675.

[12]

M. Ratner, The rate of mixing for geodesic and horocycle flow, Ergodic Theory Dyn. Syst., 7 (1987), 267-288. doi: 10.1017/S0143385700004004.

[13]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101 (1990), 449-482. doi: 10.1007/BF01231511.

[14]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., 165 (1990), 229-309. doi: 10.1007/BF02391906.

[15]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357.

[16]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.

[17]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618. doi: 10.1016/j.matpur.2014.07.004.

[18]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574.

[19]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732. doi: 10.1215/S0012-7094-94-07521-2.

[20]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328. doi: 10.3934/jmd.2013.7.291.

[21]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, arXiv:1501.05228.

[22]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.

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