Genericity on submanifolds and application to universal hitting time statistics

We investigate Birkhoff genericity on submanifolds of homogeneous space $X = SL_d (\mathbb{R}) \ltimes (\mathbb{R}^d)^k / SL_d (\mathbb{Z}) \ltimes (\mathbb{Z}^d)^k$, where $d \geq 2$ and $k \geq 1$ are fixed integers. The submanifolds we consider are parameterized by unstable horospherical subgroup $U$ of a diagonal flow $a_t$ in $SL_d (\mathbb{R})$. As long as the intersection of the submanifold with any affine rational subspace has Lebesgue measure zero, we show that the trajectory of $a_t$ along Lebesgue almost every point on the submanifold gets equidistributed on $X$. This generalizes the previous work of Frączek, Shi and Ulcigrai in [8]. Following the scheme developed by Dettmann, Marklof and Strömbergsson in [3], we then deduce an application to universal hitting time statistics for integrable flows.

Keywords

homogeneous dynamics, ergodic theory, equidistribution, diagonal flow

2010 Mathematics Subject Classification

Primary 37A17. Secondary 37A50, 37J35.

Full Text (PDF format)

Received 29 April 2022

Received revised 28 July 2022

Accepted 11 August 2022

Published 7 April 2023