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Pure and Applied Mathematics Quarterly
Volume 19 (2023)
Number 2
Genericity on submanifolds and application to universal hitting time statistics
Pages: 529 – 573
DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a5
Author
Abstract
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.
Received 29 April 2022
Received revised 28 July 2022
Accepted 11 August 2022
Published 7 April 2023