Diffeomorphisms lying in one-parameter groups and extension of stratified homeomorphisms

In this note we first show, using results of McDuff, that for a C-r-manifol d S, diffeomorphic to the interior of a compact manifold with boundary, the class of all C-r-diffeomorphisms lying in a one parameter group, of S gene rates the connected component of is in Diff(r)(S,S), Then we use this resul t to obtain two extension theorems for stratified maps defined on some stra ta of a stratified. space X. Our extension theorems hold for Mather's abstr act stratified sets, for Whitney (b)-regular, Bekka (c)-regular Verdier (w) -regular and Lipschitz-regular stratified spaces.