Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$
Cheng Zheng
Journal of Modern Dynamics 2016, 10(02): 1-21 doi: 10.3934/jmd.2016.10.1
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.

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