Journal of Modern Dynamics (JMD)

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

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

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Cheng Zheng - Department of Mathematics, The Ohio State University, 231 W. 18th Ave., MA 350, Columbus, OH 43210, United States (email)

Abstract: 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].

Keywords:  Sparse equidistribution, geodesic excursions into cusps.
Mathematics Subject Classification:  Primary: 22E40; Secondary: 37A17.

Received: May 2015;      Revised: October 2015;      Available Online: February 2016.