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This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistenceof normally hyperbolic laminations and the theorem of Robinson on thestructural stability of diffeomorphisms that satisfy Axiom A and the strongtransversality condition (SA). We generalize these results by introducing ageometric object: the stratification of laminations. It is a stratificationwhose strata are laminations. Our main theorem implies the persistence of somestratifications whose strata are normally expanded. The dynamics is a$C^r$-endomorphism of a manifold (which is possibly not invertible). Thepersistence means that for any $C^r$-perturbation of the dynamics, there existsa close $C^r$-stratification preserved by the perturbation. This theorem in itselementary 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 thepersistence, as stratifications, of submanifolds with boundary or cornersnormally expanded. Moreover, we remark that SA diffeomorphism gives a canonicalstratifications: the stratification whose strata are the stable sets of basicpieces of the spectral decomposition. Our Main theorem then implies thepersistence of some ``normally SA'' laminations which are not normallyhyperbolic.

Persistence of stratification of normally expanded laminations