On the dimension drop conjecture for diagonal flows on the space of lattices

Abstract: Consider the set of points in a homogeneous space X=G/Gamma whose g_t orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when it has real rank 1. In this talk we will prove the conjecture for probably the most important example of the higher rank case, namely: G=SL(m+n, R), Gamma=SL(m+n,Z), and g_t = diag(exp(t/m), …, exp(t/m), exp(-t/n), …, exp(-t/n)). We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar