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Entropy and Escape of Mass in Non-Compact Homogeneous Spaces

Kadyrov, Shirali
http://rave.ohiolink.edu/etdc/view?acc_num=osu1272027404

Abstract Details

2010, Doctor of Philosophy, Ohio State University, Mathematics.
We study the limit of a sequence of probability measures on a non-compact homogeneous spaces invariant under diagonalizable flow. In this context the limit measure may not be probability. Our particular interest is to study how much mass could be left in the limit if we additionally assume that our measures have high entropy. This is a part of the project on generalizing a theorem of M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh. They prove that for any sequence (μi) of probability measure on SL2(ℤ)\SL2(ℝ) invariant under the time-one-map T of geodesic flow with entropies hμi(T) ≥ c one has that any weak* limit μ of μi has at least μ(X) ≥ 2c−1 mass left. We first consider the homogeneous space SL3(ℤ)\SL3(ℝ) with an action T of a particular diagonal element diag(e1/2, e1/2, e-1) and prove a generalization. Next, by constructing T-invariant probability measure with high entropy we show that our result is sharp. We also consider the Hilbert Modular space type quotient spaces and again obtain the a generalization by studying any diagonal element. As an application one can calculate an upper bound for the Hausdorff dimension of the set of points that lie on divergent trajectories with respect to the diagonal element considered, giving an alternative proof to a result of Y. Cheung. The work regarding SL3(ℤ)\SL3(ℝ) is joint work with my co-adviser M.Einsiedler.
Nimish Shah (Advisor)
Vitaly Bergelson (Committee Member)
Alexander Leibman (Committee Member)
Christopher Jekeli (Committee Member)
104 p.
Mathematics
Ergodic Theory; Homogeneous spaces; entropy

Recommended Citations

Citations

  • Kadyrov, S. (2010). Entropy and Escape of Mass in Non-Compact Homogeneous Spaces [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1272027404

    APA Style (7th edition)

  • Kadyrov, Shirali. Entropy and Escape of Mass in Non-Compact Homogeneous Spaces. 2010. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1272027404.

    MLA Style (8th edition)

  • Kadyrov, Shirali. "Entropy and Escape of Mass in Non-Compact Homogeneous Spaces." Doctoral dissertation, Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1272027404

    Chicago Manual of Style (17th edition)

osu1272027404
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This open access ETD is published by The Ohio State University and OhioLINK.