Dynamics of Unipotent Subgroups on Infinite Volume Space / Minju Lee.

9798534652192

[New Haven, Connecticut] : Yale University, 2021.

1 online resource (299 pages)

This thesis consists of five separate projects. They are organized into the following sections:1. Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends.In joint work with Oh, we establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in SO(d, 1) acting on the space Γ\\SO(d, 1), assuming that the associated hyperbolic manifold M = Γ\\Hd is a convex cocompact manifold with Fuchsian ends.2. Topological proof of Benoist-Quint.Let G = SO°(d,1), Δ < G be a cocompact lattice, and Γ < G be a Zariski-dense discrete subgroup. In joint work with Oh, we prove that every Γ-orbit in Δ\\G is either finite or dense, which is an earlier theorem of Benoist-Quint. While Benoist and Quint's proof is based on their classification results for stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on Γ\\G.3. Anosov subgroups: local mixing, counting, and equidistribution.In joint work with Edwards and Oh, we obtain the asymptotic behavior of matrix coefficients Φ(g) = (gf1,f2) in L2(Γ\\G) for local functions [formula omitted], where G is a connected semisimple real linear Lie group and Γ , G is an Anosov subgroup. As an application, for a symmetric subgroup H of G, we obtain a counting result for 1) Γ-orbits in the bisector of generalized Cartan decomposition of G, as well as 2) discrete Γ-orbits in H\\G, analogous to Duke-Rudnick-Sarnak and Eskin-McMullen.4. Invariant measures for horospherical actions and Anosov groups. Let G be a connected semisimple real linear Lie group, and Γ < G is an Anosov subgroup. For a maximal horospherical subgroup N of G, let A be a maximal real split torus and M be a maximal compact subgroup which normalizes N. In joint work with Oh, we prove that the space of all nontrivial NM-invariant, ergodic, and A-quasi-invariant measures on Γ\\G up to proportionality, is homeomorphic to RrankG-1.5. Ergodic decompositions of geometric measures on Anosov homogeneous spaces. Let Γ\\G be an Anosov subgroup of a connected semisimple real linear Lie group G. For a maximal horospherical subgroup N of G, we show that the space of all non-trivial NM-invariant ergodic and A-quasi-invariant Radon measures on Γ\\G, up to proportionality, is homeomorphic to [formula omitted], where A is a maximal real split torus and M is a maximal compact subgroup which normalizes N. One of the main ingredients is to establish the NM-ergodicity of all Burger-Roblin measures.

Proquest dissertation Dissertations & Theses @ Yale University.

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English

December 16, 2021

Ph.D. Yale University 2021.

Academic theses.