Date of Award
Spring 2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Oh, Hee
Abstract
In this thesis, we prove some dynamical and geometric properties of diagonal flows on \emph{Anosov homogeneous spaces}, generalizing classical results for the geodesic flow on closed hyperbolic surfaces. Let $G$ be a connected semisimple real algebraic group and $\Gamma$ be a Zariski dense Anosov subgroup of $G$. On the Anosov homogeneous space $\Gamma \backslash G$, we consider a \emph{one-parameter diagonal flow} $a_{t\v}$ ($t \in \R$) parameterized by a family $D_\Gamma \cong \R^{\rankG-1}$ of directions $\v$. We prove that $a_{t\v}$ is \emph{locally mixing} with respect to the \emph{Bowen--Margulis--Sullivan measure} $\BMS$ with recurrence rate proportional to $1/t^{(\rankG-1)/2}$. More generally, our results apply to perturbed flows $a_{t\v+\sqrt{t}\u}$ for certain $\u$ transverse to $\v$ and we obtain bounds for the associated correlation functions. This generality turns out to be important for applications to equidistribution problems in $\Gamma \backslash G$. In his 1970 thesis, Margulis used mixing of the geodesic flow to study the distribution of closed geodesics in the unit tangent bundle of a compact hyperbolic manifold and obtain his celebrated Prime Geodesic Theorem. A later work of Margulis--Mohammadi--Oh generalized this approach to prove \emph{joint equidistribution} of closed geodesics and \emph{holonomies} for convex cocompact rank one manifolds. In the Anosov homogeneous space $\Gamma \backslash G$, \emph{cylinders} homeomorphic to $\mathbb{S}^1 \times \R^{\rankG-1}$ are the analogues to closed geodesics. Generalizing the techniques of the previous works, we prove joint equidistribution of cylinders and holonomies. Viewing the \emph{periods} of cylinders as a vector in $\LieA \cong \R^{\rankG}$, we can prove equidistribution of cylinders with periods lying in special families of growing subsets of $\LieA$. Using this, we also prove multiple correlations of Jordan (and Cartan) spectra for $d$-tuples of Anosov representations. This novel viewpoint gives a vast generalization of a theorem due to Schwartz--Sharp on correlations of length spectra for a pair of hyperbolic structures.
Recommended Citation
Chow, Michael, "Dynamics of Diagonal Flows in Anosov Homogeneous Spaces" (2024). Yale Graduate School of Arts and Sciences Dissertations. 1477.
https://elischolar.library.yale.edu/gsas_dissertations/1477
