Persistence of stratifications of normally expanded laminations
Mémoires de la Société Mathématique de France, no. 134 (2013) , 113 p.
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This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistence of normally hyperbolic laminations and implies several structural stability theorems. We generalize the concept of lamination by defining a new object: the stratification of laminations. It is a stratification whose strata are laminations. The 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 and with critical points). The persistence means that any C r -perturbation of the dynamics preserves a C r -close stratification. If the stratification consists of a single stratum, the main theorem implies the persistence of normally expanded laminations by endomorphisms, and hence implies HPS theorem. Another application of this theorem is the persistence, as stratifications, of submanifolds with boundary or corners normally expanded. Several examples are also given in product dynamics. As diffeomorphisms that satisfy axiom A and the strong transversality condition (AS) defines canonically two stratifications of laminations: the stratification whose strata are the (un)stable sets of basic pieces of the spectral decomposition. The main theorem implies the persistence of some “normally AS” laminations which are not normally hyperbolic and other structural stability theorems.

Ce travail s’inscrit dans le prolongement de celui de Hirsch-Pugh-Shub (HPS) sur la persistance des laminations normalement hyperboliques, et implique plusieurs théorèmes de stabilité structurelle. On généralise le concepte de lamination par une nouvelle catégorie d’objets : les stratifications de laminations. Il s’agit de stratifications, dont les strates sont des laminations. On propose alors un théorème assurant la persistance de certaines stratifications dont chaque strate est une lamination normalement dilatée. La dynamique est un C r -endomorphisme d’une variété (qui n’est donc pas forcément inversible et qui peut avoir des points critiques). La persistance signifie que toute C r -perturbation de la dynamique préserve une stratification C r -proche. Quand la stratification est formée d’une unique strate, le théoreme principal donne la persistance des laminations normalement dilatées par un endomorphisme, et implique ainsi le théorème de HPS. Une autre application de ce théorème est la persistance des variétés à bord ou à coins normalement dilatés. Beaucoup examples sont donnés facilement en dynamique produit. Aussi les difféomorphismes vérifiant l’axiome A et la condition de transversalité forte (ATF) possèdent deux stratifications de laminations canoniques : celle dont les strates sont les ensembles stables (resp. instables) de ses pièces basiques. Ainsi, notre théorème implique la persistance de certaines laminations “normalement ATF” qui ne sont pas normalement hyperboliques et d’autres théorèmes de stabilité structurelle.

DOI: 10.24033/msmf.444
Keywords: Laminations, Stratifications, Structural Stability, Persistence, Hyperbolic Dynamics, Endomorphisms, Axiom A, Product dynamics.
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Berger, Pierre. Persistence of stratifications of normally expanded laminations. Mémoires de la Société Mathématique de France, Serie 2, , no. 134 (2013), 113 p. doi : 10.24033/msmf.444. http://numdam.org/item/MSMF_2013_2_134__1_0/

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