Winning sets of Diophantine measured foliations
Howard Masur (University of Chicago)
Abstract: In the 1960's W.Schmidt invented a game now called a Schmidt game to
be played in R^n. Associated are what are called winning sets which have
various nice properties; one of which is full Hausdorff dimension. The main
motivating example of a winning set which Schmidt considered is the subset of
reals with bounded continued fraction expansion. Classically these are the
reals that are badly approximable by fractions. They also correspond in the
moduli space H^2/SL(2,Z) to hyperbolic geodesics that stay in a compact set.
One can formulate a similar condition for a measured foliation on a higher
genus surfaces to be badly
approximated by simple closed curves. These correspond to Teichmuller geodesics
that stay in a compact subset of the corresponding moduli space. These are
called Diophantine foliations. After giving the background on winning sets I
will discuss the theorem, joint with Jon Chaika and Yitwah Cheung that the set
of Diophantine foliations is Schmidt winning as a subset of PMF, Thurston's
sphere of measured foliations.