Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

Nimish Shah; Lei Yang

doi:10.3934/dcds.2020227

Article Contents

2020, Volume 40, Issue 9: 5247-5287. Doi: 10.3934/dcds.2020227

This issue Previous Article A topological study of planar vector field singularities Next Article Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

Nimish Shah^1, and
Lei Yang^2, ,

1.
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
2.
College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

^* Corresponding author: Lei Yang
^* Corresponding author: Lei Yang

Received: September 2019

Revised: February 2020

Published: June 2020

This material is based upon work supported by the National Science Foundation under Grant Nos. 1301715 and 1700394.
The second author is supported in part by ISF grant 2095/15, ERC grant AdG 267259, NSFC grant 11743006, 11801384 and the Fundamental Research Funds for the Central Universities YJ201769

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Abstract

In this paper, we study an analytic curve $ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $ in the space of $ m $ by $ n $ real matrices, and show that if $ \varphi $ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space $ G/\Gamma $, and prove that under the action of some expanding diagonal subgroup $ A = \{a(t): t \in \mathbb{R}\} $, the translates of the curve tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. The proof relies on the linearization technique and representation theory.

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Mathematics Subject Classification: Primary: 11J13, 11J83; Secondary: 22E40, 37A17.

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