Geometry, Topology and Dynamics Seminar
Yitwah Cheung
San Francisco State
Nonergodic Directions in a Rational Billiard
Abstract: Consider billiards in a polygon whose angles are
rational multiples of $\pi$. By a theorem of Kerchkoff-Masur-Smillie,
the translation flow is uniquely ergodic in almost every direction.
This means that for almost every direction $\theta$ in the unit circle,
every billiard trajectory with initial direction $\theta$ will be uniformly distributed inside the polygon. Much work has been done
to understand the nature of the set of exceptional "nonergodic"
directions. In this talk, I will describe some of the ideas that
go into the proof of the following "dichotomy" result:
Let $P_\lambda$ be the 2-by-1 rectangular table with a wall of
length $\lambda$ inserted orthogonally at midpoint of a longer edge.
The Hausdorff dimension of the set nonergodic directions in $P_\lambda$
can only be $0$ or $1/2$, with the latter happening
if and only if the continued fraction expansion of $\lambda$ satisfies the Perez-Marco condition. (This condition
appears in the linearisation problem in the study of holomorphic
dynamics near a fixed point.)
This result is joint with Pascal Hubert and Howard Masur.
Monday November 10, 2008 at 3:00 PM in SEO 612