S. G. Dani

On orbits of unipotent flows on homogeneous spaces

We show that if ( u t ) is a one-parameter subgroup of SL ( n , ℝ) consisting of unipotent matrices, then for any e > 0 there exists a compact subset K of SL( n , ℝ)/SL( n , ℤ) such that the following holds: for any g ∈ SL( n , ℝ) either m ({ t ∈ [0, T ] | u t g SL ( n , ℤ) ∈ K }) > (1 – e) T for all large T ( m being the Lebesgue measure) or there exists a non-trivial ( g −1 u t g )-invariant subspace defined by rational equations. Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U , which is not contained in any proper closed subgroup of G , we have G = DΓ U . The decomposition is applied to get results on Diophantine approximation.

Anosov automorphisms on compact nilmanifolds associated with graphs

We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N. We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/F to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every n > 17 there exist a n-dimensional 2-step simply connected nilpotent Lie group N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ in N such that N/F admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups N of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

On orbits of endomorphisms of tori and the Schmidt game

We show that there exists a subset F of the n-dimensional torus T n such that F has Hausdorff dimension n and for any xe F and any semisimple automorphism σ of T n the closure of the σ -orbit of x contains no periodic points.

Contraction subgroups and semistable measures on p-adic lie groups

for all t ^ 0, for a fixed ce (0,1), have attractedconsiderable attention of various researchers in recent years (cf. [3], [5] and otherreferences cited therein). A detailed study of semistable measures on (real) Lie groupsis carried out in [5]. In this context it is of interest to study semistable measures onthe class of p-adic Lie groups, which is another significant class of locally compactgroups.It is known for a general metrizable locally compact group G (cf. [4], proposition4-2) that if {ji

On automorphism groups of connected Lie groups

AbstractWe prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of

Invariance groups and convergence of types of measures on lie groups

GL(\mathcal{G})

On invariant measures of the Euclidean algorithm

, where