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Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication

Published online by Cambridge University Press:  16 April 2012

CH. BONATTI ,
S. CROVISIER ,
L. J. DÍAZ  and
N. GOURMELON
CH. BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne, BP 47 870, 1078 Dijon Cedex, France (email: bonatti@u-bourgogne.fr)
S. CROVISIER
Affiliation:
Institut Galilée, Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France (email: crovisie@math.univ-paris13.fr)
L. J. DÍAZ
Affiliation:
Departamento de Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro RJ, Brazil (email: lodiaz@mat.puc-rio.br)
N. GOURMELON
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, cours de la Libération, 33405 Talance Cedex, France (email: nicholas.gourmelon@math.u-bordeaux1.fr)
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Abstract

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013 , pp. 739 - 776
Copyright
Copyright © 2012 Cambridge University Press

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