Genericity on submanifolds and application to Universal hitting time statistics

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.