Genericity on submanifolds and application to Universal hitting time statistics

TL;DR

This paper proves Birkhoff genericity for certain submanifolds in homogeneous spaces, leading to new results on equidistribution and universal hitting time statistics for integrable flows.

Contribution

It extends Birkhoff genericity results to submanifolds parameterized by unstable horospherical groups, generalizing previous work and applying to hitting time statistics.

Findings

Almost every point's trajectory is equidistributed on the space.

Generalizes previous genericity results to broader submanifolds.

Provides applications to universal hitting time statistics.

Abstract

We investigate Birkhoff genericity on submanifolds of homogeneous space $X=SL_d(\bR)\ltimes (\bR^d)^k/ SL_d(\bZ)\ltimes (\bZ^d)^k$, 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 $SL_d(\bR)$. 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 Fr\k{a}czek, Shi and Ulcigrai in \cite{Shi_Ulcigrai_Genericity_on_curves_2018}. Following the scheme developed by Dettmann, Marklof and Str\"{o}mbergsson in \cite{Marklof_Universal_hitting_time_2017}, we then deduce an application to universal hitting time statistics for integrable flows.