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Structural stability of attractor–repellor endomorphisms with singularities

Published online by Cambridge University Press:  10 June 2011

PIERRE BERGER
PIERRE BERGER*
Affiliation:
CNRS-IMPA-LAGA UMR 7539, Institut Galilée Université, Paris 13, 99 avenue, J.B. Clément, 93430 Villetaneuse, France (email: berger@math.univ-paris13.fr)
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Abstract

We prove a theorem on the structural stability of smooth attractor–repellor endomorphisms of compact manifolds, with singularities. By attractor–repellor, we mean that the non-wandering set of the dynamics f is the disjoint union of an expanding compact subset with a hyperbolic attractor on which f acts bijectively. The statement of this result is both infinitesimal and dynamical. To our knowledge, this is the first in this hybrid direction. Our results also generalize Mather’s theorem in singularity theory, which states that infinitesimal stability implies structural stability for composed mappings to the larger category of laminations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 1 - 33
Copyright
Copyright © Cambridge University Press 2011

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