Best possible rates of distribution of dense lattice orbits in homogeneous spaces

The Distribution of Periodic Torus Orbits on Homogeneous Spaces

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up ...

Distribution of Periodic Torus Orbits on Homogeneous Spaces

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up ...

Nondense Orbits of Flows on Homogeneous Spaces

Let F be a nonquasiunipotent one-parameter (cyclic) subgroup of a unimodular Lie group G, Γ a discrete subgroup of G. We prove that for certain classes of subsets Z of the homogeneous space G/Γ, the set of points in G/Γ with F -orbits staying away from Z has full Hausdorff dimension. From this we derive applications to geodesic flows on manifolds of constant negative curvature.

Nearly Countable Dense Homogeneous Spaces

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

Homogeneous Twistor Spaces And Nilpotent Orbits