**Memoires de la Societe Mathematique de France**

Volume: 134;
2013;
113 pp;
Softcover

MSC: Primary 37; 34; 57;
**Print ISBN: 978-2-85629-767-4
Product Code: SMFMEM/134**

List Price: $45.00

AMS Member Price: $36.00

# Persistence of Stratifications of Normally Expanded Laminations

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*Pierre Berger*

A publication of the Société Mathématique de France

This manuscript complements the Hirsch-Pugh-Shub (HPS) theory
on persistence of normally hyperbolic laminations and implies several
structural stability theorems.

The author generalizes 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 the 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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in laminations.