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Genericity on Submanifolds and Equidistribution of Polynomial Trajectories on Homogeneous Spaces

Zhang, Han
http://orcid.org/0000-0002-1637-3894
http://rave.ohiolink.edu/etdc/view?acc_num=osu1618231544346722

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
Homogeneous dynamics is the study of dynamics of various flows on homogeneous spaces. It has far-reaching impacts on other mathematical fields. Many applications of homogeneous dynamics is achieved via equidistribution of certain trajectories on homogeneous spaces. This work is comprised of two main parts. In the first part, we investigate Birkhoff genericity on certain submanifold of $X=SL_d(\bR)\ltimes (\bR^d)^k/ SL_d(\bZ)\ltimes (\bZ^d)^k$, where $d\geq 2$ and $k\geq 1$ are fixed integers. The submanifold we consider is parameterized by unstable horospherical subgroup $U$ of a diagonal flow $a_t$ in $SL_d(\bR)$. Under the assumption that the intersection of the submanifold with affine rational subspaces 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. Following the scheme developed by Dettmann, Marklof and Str\"{o}mbergsson, we then deduce an application of our results to universal hitting time statistics for integrable flows. In the second part, we study the limit distribution of $k$-dimensional polynomial trajectories on homogeneous spaces, where $k\geq 2$ is a fixed integer. When the averaging is taken on certain expanding boxes on $\bR^k$ and assume certain conditions on polynomial trajectories, we generalize Shah's limit distribution theorem of polynomial trajectories to higher dimensional case.
Nimish Shah (Advisor)
James Cogdell (Committee Member)
Daniel Thompson (Committee Member)
144 p.
Mathematics
Birkhoff genericity, submanifolds, height functions, polynomial trajectories, linearization technique

Recommended Citations

Citations

  • Zhang, H. (2021). Genericity on Submanifolds and Equidistribution of Polynomial Trajectories on Homogeneous Spaces [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618231544346722

    APA Style (7th edition)

  • Zhang, Han. Genericity on Submanifolds and Equidistribution of Polynomial Trajectories on Homogeneous Spaces. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1618231544346722.

    MLA Style (8th edition)

  • Zhang, Han. "Genericity on Submanifolds and Equidistribution of Polynomial Trajectories on Homogeneous Spaces." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618231544346722

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.