In this note we first show, using results of McDuff, that for a Cr-manifold S,
diffeomorphic to the interior of a compact manifold with boundary, the class of all
Cr-diffeomorphisms lying in a one parameter group of S generates the connected
component of 1S in Diffr (S, S).
Then we use this result to obtain two extension
theorems for stratified maps defined on some strata of a stratified space [Xscr ]. Our
extension theorems hold for Mather's abstract stratified sets, for Whitney (b)-regular,
Bekka (c)-regular, Verdier (w)-regular and Lipschitz-regular stratified spaces.