EVENT DETAILS AND ABSTRACT

Title: Divergent trajectories of homogeneous flows
Speaker: Professor Yitwah Cheung
Speaker Info: San Francisco State University
Brief Description:
Special Note:
Abstract:

Let G be a connected, semisimple Lie group, Gamma a lattice in G, and F={g_t} a one-parameter subgroup such that Ad g_1 has an eigenvalue of absolute value not equal to one. Assume G/Gamma is noncompact and consider the flow on G/Gamma induced by F acting by left multiplication. A trajectory of F is said to be bounded if its closure in G/Gamma is compact; it is divergent if it eventually leaves every compact subset of G/Gamma. Let BD(F) (resp. DV(F)) denote the union of all bounded (resp. divergent) trajectories. By ergodicity of the flow, both of these sets have measure zero. Kleinbock-Margulis showed that BD(F)always has Hausdorff dimension equal to dim G. A corollary of this result is Schmidt's theorem that the set of badly approximable vectors in R^d has Hausdorff dimension d. Much less is known about the Hausdorff dimension of DV(F) in this setting, except in the case of a rank one lattice.

In this talk, we consider the Hausdorff dimension of DV(F) in the special case when G=SL(3,R), Gamma=SL(3,Z) and g_t=diag(e^t,e^t,e^{-2t}). We show that the Hausdorff dimension of DV(F) is equal to dim G - 2/3. (For the theory of simultaneous Diophantine approximation, the implication is the Hausdorff dimension of the set of singular vectors in the plane is 4/3.) The proof involves a multi-dimensional generalization of the theory of continued fractions from the perspective of the best approximation properties of convergents. As an application of these ideas, we also show there are divergent trajectories that exit to infinity at an arbitrarily slow prescribed rate, answering a question of A.N. Starkov.