  The theorems of M. Ratner, describing the finite ergodic invariant measures
and the orbit closures for unipotent flows on homogeneous spaces of Lie groups,
are extended for actions of subgroups generated by unipotent elements. More
precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed
subgroup of G. Let W be a subgroup of G such that Ad(W) is contained in the
Zariski closure (in the group of automorphisms of the Lie algebra of G) of the
subgroup generated by the unipotent elements of Ad(W). Then any finite ergodic
invariant measure for the action of W on G/Gamma is a homogeneous measure
(i.e., it is supported on a closed orbit of a subgroup preserving the measure).
Moreover, if G/Gamma has finite volume (i.e., has a finite G-invariant
measure), then the closure of any orbit of W on G/Gamma is a homogeneous set
(i.e., a finite volume closed orbit of a subgroup containing W). Both the above
results hold if W is replaced by any subgroup Lambda of W such that W/Lambda
has finite volume.
