Geometry, Topology and Dynamics Seminar
Yitwah Cheung
San Francisco State
Hausdorff dimension of the set of Singular Vectors
Abstract: Singular vectors in $\mathbb R^d$ correspond to divergent trajectories of the
homogeneous flow on $SL(d+1,\mathbb R)/SL(d+1,\mathbb Z)$ induced by the one parameter
subgroup $\mathrm{diag}(e^t,...,e^t,e^{-dt})$ acting by left multiplication. In this
talk, I will sketch a proof of the following result: the Hausdorff dimension of
the set of singular vectors in $\mathbb R^2$ is $4/3$. (Alternatively, the set of points
lying on divergent trajectories of the homogeneous flow on $SL(3,\mathbb R)/SL(3,\mathbb Z)$ has
Hausdorff dimension $22/3$.) The main idea involves a multi-dimensional
generalisation of continued fraction theory from the perspective of the best
approximation properties of convergents. As an application, we answer a
question of A.N. Starkov regarding the existence of slowly divergent
trajectories.
Monday December 3, 2007 at 3:00 PM in SEO 512