  This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistence
of normally hyperbolic laminations and the theorem of Robinson on the
structural stability of diffeomorphisms that satisfy Axiom A and the strong
transversality condition (SA). We generalize these results by introducing a
geometric object: the stratification of laminations. It is a stratification
whose strata are laminations. Our main theorem implies the persistence of some
stratifications whose strata are normally expanded. The dynamics is a
$C^r$-endomorphism of a manifold (which is possibly not invertible). The
persistence means that for any $C^r$-perturbation of the dynamics, there exists
a close $C^r$-stratification preserved by the perturbation. This theorem in its
elementary statement (the stratification is constituted by a unique stratum)
gives the persistence of normally expanded laminations by endomorphisms,
generalizing HPS theory. Another application of this theorem is the
persistence, as stratifications, of submanifolds with boundary or corners
normally expanded. Moreover, we remark that SA diffeomorphism gives a canonical
stratifications: the stratification whose strata are the stable sets of basic
pieces of the spectral decomposition. Our Main theorem then implies the
persistence of some ``normally SA'' laminations which are not normally
hyperbolic.
