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Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

Part of: Diophantine equations Ergodic theory

Published online by Cambridge University Press:  22 December 2021

Runlin Zhang
Runlin Zhang*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, 1000871Beijing, PR Chinazhangrunlinmath@outlook.com
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Abstract

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$, let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$-subgroup. There is a $H$-invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$. Given a sequence $(g_n)$ in $G$, we study the limiting behavior of $(g_n)_*\mu _H$ under the weak-$*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$. We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

Keywords

homogeneous dynamicsintegral pointsequidistribution

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 12 , December 2021 , pp. 2657 - 2698
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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