Genericity on submanifolds and application to Universal hitting time statistics

DOI10.4310/PAMQ.2023.V19.N2.A5arXiv2105.11068MaRDI QIDQ6368350

Han Zhang

Publication date: 23 May 2021

Abstract: We investigate Birkhoff genericity on submanifolds of homogeneous space $$, where $dgeq2$ and $kgeq1$ are fixed integers. The submanifolds we consider are parameterized by unstable horospherical subgroup $U$ of a diagonal flow $a_t$ in $$. 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 Frk{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.

Mathematics Subject Classification ID

Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Dynamical systems and their relations with probability theory and stochastic processes (37A50) Homogeneous flows (37A17) Dynamical systems with singularities (billiards, etc.) (37C83)

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