arXiv subjects

C. Davis Buenger

Publications and source records attributed to C. Davis Buenger.

2 recordsLinked to original sources

arXiv2015

Random Walks on Homogeneous Spaces by Sparse Solvable Measures

The paper analyzes a specific class of random walks on quotients of $X:=\text{SL}(k,{\Bbb R})/ Γ$ for a lattice $Γ$. Consider a one parameter diagonal subgroup, $\{g_t\}$, with an associated abelian expanding horosphere, $U\cong {\Bbb R}^k$, and let $ϕ:[0,1]\rightarrow U$ be a sufficiently smooth curve satisfying the condition that that the derivative of $ϕ$ spends $0$ time in any one subspace of ${\Bbb R}^k$. Let $ μ_U$ be the measure defined as $ϕ_*λ_{[0,1]},$ where $λ_{[0,1]}$ is the Lebesgue measure on $[0,1]$. Let $μ_A$ be a measure on the full diagonal subgroup of $\text{SL}(k,{\Bbb R})$, such that almost surely the random walk on the diagonal subgroup $A$ with respect to this measure grows exponentially in the direction of the cone expanding $U$. Then the random walk starting at any point $z\in X$, and alternating steps given by $μ_U$ and $μ_A$ equidistributes respect to $\text{SL}(k,{\Bbb R})$-invariant measure on $X$. Furthermore, the measure defined by $μ_A*μ_U*\dots*μ_A* μ_U*δ_z$ converges exponentially fast to the $\text{SL}(k,{\Bbb R})$-invariant measure on $X$.

math.DS↗

arXiv2014

Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

Let $G$ be a semisimple Lie group of rank $1$ and $Γ$ be a torsion free discrete subgroup of $G$. We show that in $G/Γ$, given $ε>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $ δ$ for $1-ε$ proportion of the time for some $δ>0$. The result also holds for any finitely generated discrete subgroup $Γ$ and this generalizes Dani's quantitative nondivergence theorem \cite{D} for lattices of rank one semisimple groups. Furthermore, for a fixed $ε>0$ there exists an injectivity radius $δ$ such that for any unipotent trajectory $\{u_tx\}_{t\in [0,T]}$, either it spends at least $1-ε$ proportion of the time in the set with injectivity radius larger than $δ$ for all large $T>0$ or there exists a $\{u_t\}_{t\in\mathbb{R}}$-normalized abelian subgroup $L$ of $G$ which intersects $gΓg^{-1}$ in a small covolume lattice. We also extend these results when $G$ is the product of rank-$1$ semisimple groups and $Γ$ a discrete subgroup of $G$ whose projection onto each nontrivial factor is torsion free.

math.DS↗