Semi-differentiable stratified morphisms

Near each stratum of a stratified space we use canonical distributions to i mprove known theorems on continuous controlled lifting of vector fields. We define semidifferentiable and horizontally-C-1 stratified controlled morph isms f : X --> X' between stratified spaces. When X' is a smooth manifold, f is semi-differentiable. Otherwise semi-differentiability is equivalent to having bounded derivatives and being horizontally-C-1. A semi-differentiab le stratified homeomorphism preserves regularity of stratified subspaces. ( C) Academie des Sciences/Elsevier, Paris.