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Non-divergence of unipotent flows on quotients of rank-one semisimple groups

Published online by Cambridge University Press:  28 December 2015

C. DAVIS BUENGER  and
CHENG ZHENG
C. DAVIS BUENGER
Affiliation:
100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210-1174, USA email buenger@math.ohio-state.edu, czheng@math.ohio-state.edu
CHENG ZHENG
Affiliation:
100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210-1174, USA email buenger@math.ohio-state.edu, czheng@math.ohio-state.edu
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Abstract

Let $G$ be a semisimple Lie group of rank one and $\unicode[STIX]{x1D6E4}$ be a torsion-free discrete subgroup of $G$. We show that in $G/\unicode[STIX]{x1D6E4}$, given $\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $\unicode[STIX]{x1D6FF}$ for a $1-\unicode[STIX]{x1D716}$ proportion of the time, for some $\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup $\unicode[STIX]{x1D6E4}$ and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed $\unicode[STIX]{x1D716}>0$, there exists an injectivity radius $\unicode[STIX]{x1D6FF}$ such that, for any unipotent trajectory $\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a $1-\unicode[STIX]{x1D716}$ proportion of the time in the set with injectivity radius larger than $\unicode[STIX]{x1D6FF}$, for all large $T>0$, or there exists a $\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup $L$ of $G$ which intersects $g\unicode[STIX]{x1D6E4}g^{-1}$ in a small covolume lattice. We also extend these results to when $G$ is the product of rank-one semisimple groups and $\unicode[STIX]{x1D6E4}$ a discrete subgroup of $G$ whose projection onto each non-trivial factor is torsion free.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 103 - 128
Copyright
© Cambridge University Press, 2015 

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