On topological conjugacy of left invariant flows on semisimple and affine Lie groups

in this paper, we study the flows of nonzero left invariant vector fields on lie groups with respect to topological conjugacy. using the fundamental domain method, we are able to show that on a simply connected nilpotent lie group any such flows are topologically conjugate. combining this result with the iwasawa decomposition, we find that on a noncompact semisimple lie group the flows of two nilpotent or abelian fields are topologically conjugate. finally, for affine groups g = hv , v e" n, we show that the conjugacy class of a left invariant vector field does not depend on its euclidean component.