Divergent trajectories on products of homogeneous spaces

Limit Distributions of Polynomial Trajectories on Homogeneous Spaces

Divergent Torus Orbits in Homogeneous Spaces of Qrank Two

Let G be a semisimple algebraic Qgroup. let. F be an arithmetic subgroup of G . and let, T be an E-split torus in G . We prove that if there is a divergent Tn-orbit in r \ G R , and Q-rank G < 2, t,hen dim T < Q-rankG. This provides a partial answer to a question of G . Tomanov and B. Weiss.

Localization operators on homogeneous spaces

Let $G$ be a locally compact group, $H$ be a compact subgroup of $G$ and $varpi$ be a representation of the homogeneous space $G/H$ on a Hilbert space $mathcal H$. For $psi in L^p(G/H), 1leq p leqinfty$, and an admissible wavelet $zeta$ for $varpi$, we define the localization operator $L_{psi,zeta} $ on $mathcal H$ and we show that it is a bounded operator. Moreover, we prove that the localizat...

Bounded Trajectories in Homogeneous Spaces of Semisimple Lie Groups

This is a report on a joint work with G.A. Margulis. The problem to be discussed is apparently motivated by number theory, more precisely, by problems in Diophantine approximation. Let us start by recalling that the set B of badly approximable real numbers has full Hausdorff dimension at any point, in other words, for any nonempty open subset W of R, dim(W ∩ B) = 1 (cf. [J,S]). On the other han...

Flows on Homogeneous Spaces

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain ows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprind zuk in 1964. We also prove several related hypotheses of Baker and Sprind zuk formulated in 1970...