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,, \emph{Invent. Math.}, 47 (1978), 101. doi: 10.1007/BF01578067. Google Scholar

[2]

S. G. Dani, Invariant measures and minimal sets of horospherical flows,, \emph{Invent. Math.}, 64 (1981), 357. doi: 10.1007/BF01389173. Google Scholar

[3]

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

[4]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

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H. Furstenberg, The unique ergodicity of the horocycle flow,, in \emph{Recent Advances in Topological Dynamics} (Proc. Conf., (1972), 95. Google Scholar

[6]

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

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in \emph{Sinai's Moscow Seminar on Dynamical Systems}, (1996), 141. Google Scholar

[8]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds,, \emph{Ann. of Math. (2)}, 148 (1998), 339. doi: 10.2307/120997. Google Scholar

[9]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, \emph{Invent. Math.}, 138 (1999), 451. doi: 10.1007/s002220050350. Google Scholar

[10]

G. A. Margulis, Discrete subgroups and ergodic theory,, in \emph{Number Theory, (1987), 377. Google Scholar

[11]

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

[12]

M. Ratner, The rate of mixing for geodesic and horocycle flow,, \emph{Ergodic Theory Dyn. Syst.}, 7 (1987), 267. doi: 10.1017/S0143385700004004. Google Scholar

[13]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups,, \emph{Invent. Math.}, 101 (1990), 449. doi: 10.1007/BF01231511. Google Scholar

[14]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups,, \emph{Acta Math.}, 165 (1990), 229. doi: 10.1007/BF02391906. Google Scholar

[15]

M. Ratner, On Raghunathan's measure conjecture,, \emph{Ann. of Math. (2)}, 134 (1991), 545. doi: 10.2307/2944357. Google Scholar

[16]

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

[17]

P. Sarnak and A. Ubis, The horocycle flow at prime times,, \emph{J. Math. Pures Appl. (9)}, 103 (2015), 575. doi: 10.1016/j.matpur.2014.07.004. Google Scholar

[18]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces,, \emph{Math. Ann.}, 289 (1991), 315. doi: 10.1007/BF01446574. Google Scholar

[19]

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

[20]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows,, \emph{J. Mod. Dyn.}, 7 (2013), 291. doi: 10.3934/jmd.2013.7.291. Google Scholar

[21]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution,, \arXiv{1501.05228}., (). Google Scholar

[22]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity,, \emph{Ann. of Math. (2)}, 172 (2010), 989. doi: 10.4007/annals.2010.172.989. Google Scholar

show all references

References:
[1]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces,, \emph{Invent. Math.}, 47 (1978), 101. doi: 10.1007/BF01578067. Google Scholar

[2]

S. G. Dani, Invariant measures and minimal sets of horospherical flows,, \emph{Invent. Math.}, 64 (1981), 357. doi: 10.1007/BF01389173. Google Scholar

[3]

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

[4]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

[5]

H. Furstenberg, The unique ergodicity of the horocycle flow,, in \emph{Recent Advances in Topological Dynamics} (Proc. Conf., (1972), 95. Google Scholar

[6]

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

[7]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in \emph{Sinai's Moscow Seminar on Dynamical Systems}, (1996), 141. Google Scholar

[8]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds,, \emph{Ann. of Math. (2)}, 148 (1998), 339. doi: 10.2307/120997. Google Scholar

[9]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, \emph{Invent. Math.}, 138 (1999), 451. doi: 10.1007/s002220050350. Google Scholar

[10]

G. A. Margulis, Discrete subgroups and ergodic theory,, in \emph{Number Theory, (1987), 377. Google Scholar

[11]

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

[12]

M. Ratner, The rate of mixing for geodesic and horocycle flow,, \emph{Ergodic Theory Dyn. Syst.}, 7 (1987), 267. doi: 10.1017/S0143385700004004. Google Scholar

[13]

M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups,, \emph{Invent. Math.}, 101 (1990), 449. doi: 10.1007/BF01231511. Google Scholar

[14]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups,, \emph{Acta Math.}, 165 (1990), 229. doi: 10.1007/BF02391906. Google Scholar

[15]

M. Ratner, On Raghunathan's measure conjecture,, \emph{Ann. of Math. (2)}, 134 (1991), 545. doi: 10.2307/2944357. Google Scholar

[16]

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

[17]

P. Sarnak and A. Ubis, The horocycle flow at prime times,, \emph{J. Math. Pures Appl. (9)}, 103 (2015), 575. doi: 10.1016/j.matpur.2014.07.004. Google Scholar

[18]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces,, \emph{Math. Ann.}, 289 (1991), 315. doi: 10.1007/BF01446574. Google Scholar

[19]

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

[20]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows,, \emph{J. Mod. Dyn.}, 7 (2013), 291. doi: 10.3934/jmd.2013.7.291. Google Scholar

[21]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution,, \arXiv{1501.05228}., (). Google Scholar

[22]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity,, \emph{Ann. of Math. (2)}, 172 (2010), 989. doi: 10.4007/annals.2010.172.989. Google Scholar

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