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Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces

Buenger, Carl D, Buenger
http://rave.ohiolink.edu/etdc/view?acc_num=osu1462800914

Abstract Details

2016, Doctor of Philosophy, Ohio State University, Mathematics.
First, we let G be a semisimple Lie group of rank 1 and Γ be a torsion free discrete subgroup of G. Jointly with Cheng Zheng, we show that in G/Γ, given ε > 0, there exists δ > 0 such that any unipotent trajectory has injectivity radius larger than δ for 1 - ε proportion of the time. The result generalizes Dani’s quantitative non- divergence theorem proved when Γ is a lattice. Furthermore, for ε > 0, there exists δ > 0 such that for any unipotent trajectory {utgΓ}t∈[0,T], either the trajectory spends has injectivity radius δ for at least 1 - ε proportion of the time or there exists a {ut}t∈R-normalized abelian subgroup L of G which intersects gΓg-1 in a small covolume lattice. We also extend these results to the case when G is the product of rank-1 semisimple groups and Γ is a discrete subgroup of G whose projection onto each nontrivial factor is torsion free. Second, using quantitative non-divergence, we proves a variation of the mixing results of Rongang Shi which generalized results of Kleinbock and Margulis. Let G be a Lie group with lattice G and H be a non-compact simple Lie group such that the action of H on G/Γ has a spectral gap. Let U be a horospherical subgroup of H and A a maximal split torus in the normalizer of U. Let A+U denote the expanding cone of A as defined by Shi. We prove effective k-equidistribution of U -slice under translates by a ∈ A+U . Third, using mixing results, we analyze specific sparse solvable random walks on X := G/Γ for G, Γ and H as above and on N/Γ for a simply connected nilpotent group N and lattice Γ. Let U be a Horospherical subgroup of H and let A be a maximal split torus in the normalizer of U. Let φ : [0,1]l → U be a C1 function satisfying a non-planar condition. Let µ0 be a absolutely continuous probability measure on [0, 1]l with continuous Radon-Nykodym derivative. Let µU := φ*µ0. Let µA be a probability measure on A with suitable mean and moment assumptions. Let µ = µA * µU . Then the µ-random walk starting at z ∈ X, equidistributes with respect to the G-invariant probability measure on X. Furthermore, µ*n * dz converges exponentially fast to the G-invariant probability measure on X. Additionally, let N be a simply connected nilpotent Lie group and G be a lattice in N. Let aˆ be an ergodic automorphism of N/G which extends to an automorphism a of N. Let φ : [0,1]l → N be a C1 function satisfying certain non-planar conditions. Let µ = a*φ*µ0. Then the same results hold in this case.
Nimish Shah, Dr. (Advisor)
Vitaly Bergelson, Dr. (Committee Member)
Daniel Thompson, Dr. (Committee Member)
Thomas Magliery, Dr. (Committee Member)
114 p.
Mathematics
Non Divergence, Random Walks, Homogeneous Spaces, Ergodic Theory, Unipotent Flows, Mixing, Decay of Matrix Coefficients

Recommended Citations

Citations

  • Buenger, Buenger, C. D. (2016). Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1462800914

    APA Style (7th edition)

  • Buenger, Buenger, Carl. Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces. 2016. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1462800914.

    MLA Style (8th edition)

  • Buenger, Buenger, Carl. "Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces." Doctoral dissertation, Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1462800914

    Chicago Manual of Style (17th edition)

osu1462800914
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This open access ETD is published by The Ohio State University and OhioLINK.