Dani’s work on dynamical systems on homogeneous spaces

We describe some of S.G.Dani’s many contributions to the theory and applications of dynamical systems on homogeneous spaces, with emphasis on unipotent flows. S.G.Dani has written over 100 papers. They explore a variety of topics, including: • flows on homogeneous spaces – unipotent dynamics – applications to Number Theory – divergent orbits – bounded orbits and Schmidt’s game – topological orbit equivalence – Anosov diffeomorphisms – entropy and other invariants • actions of locally compact groups – actions of lattices – action of AutG on the Lie group G – stabilizers of points • convolution semigroups of probability measures on a Lie group • finitely additive probability measures • Borel Density Theorem • history of Indian mathematics Most of Dani’s papers (about 60) are directly related to flows on homogeneous spaces. This survey will briefly discuss several of his important contributions in this field. Notation. Let: • G = SL(n,R) (or, for the experts, G may be any connected Lie group), • {g} be a one-parameter subgroup of G, • Γ = SL(n,Z) (or, for the experts, G may be any lattice in G), and • φt(xΓ) = gxΓ for t ∈ R and xΓ ∈ G/Γ. Then φs+t = φs ◦ φt, so φt is a flow on the homogeneous space G/Γ. 2010 Mathematics Subject Classification. Primary 37A17; Secondary 11H55, 37A45.