Horizontally-C-1 controlled stratified maps and Thom's first isotopy theorem

We state results concerning horizontally-C-1 regularity for a stratified ma p f:X --> X'. It requires the existence of local stratified (a)-regular hor izontal foliations for X and X' (a smooth version of the conjectured comple x analytic "Whitney fibering") and implies a horizontally-C-1 version of Th om's first isotopy theorem. We also consider the more general notion of F-s emi-differentiability for f, F being a foliation, and state analogous theor ems. (C) 2000 Academie des sciences/Editions scientifiques et medicales Els evier SAS.