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

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.

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Buenger, C. (2016). Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Buenger, Carl. "Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces." Electronic Thesis or Dissertation. Ohio State University, 2016. OhioLINK Electronic Theses and Dissertations Center. 23 Nov 2017.

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Buenger, Carl "Quantitative Non-Divergence, Effective Mixing, and Random Walks on Homogeneous Spaces." Electronic Thesis or Dissertation. Ohio State University, 2016. https://etd.ohiolink.edu/

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