Title: Schmidt's Game and Orbits of Toral Endomorphisms
Speaker: Ryan Broderick
Speaker Info: Northwestern University
Brief Description:
Special Note:
Abstract:
It is known that an invertible n x n integer matrix with no roots of unity as eigenvalues induces a continuous self-map of the n-torus under which almost all points have dense orbits. Nevertheless, S. G. Dani proved in 1988 that if the matrix is semisimple then the set of points whose orbit closure misses a given rational has full Hausdorff dimension. In fact, he showed that this set has the stronger property of being winning in the sense of Schmidt's game, a property which is stable under countable intersections and therefore implies full dimension of the set of points with orbit closures missing all rationals under all maps as above. I will describe the game and discuss a recent joint work with L. Fishman and D. Kleinbock in which this theorem is generalized in several directions.Date: Tuesday, September 27, 2011