Abstract. – We construct locally generic C1-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes.¶We prove that these phenomena are associated to the existence of a homoclinic class H(P,f) with two specific properties:¶– in a C1-robust way, the homoclinic class H(P,f) does not admit any dominated splitting,¶– there is a periodic point P′ homoclinically related to P such that the Jacobians of P′ and P are greater than and less than one, respectively.
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Manuscrit reĉu le 13 décembre 2000.
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ID="*"This paper was partially supported by CNPq, Faperj, and Pronex Dynamical Systems (Brazil), PICS-CNRS and the Agreement Brazil-France in Mathematics. The authors acknowledge to IMPA and Laboratoire de Topologie, Université de Bourgogne, for the warm hospitality during their visits while preparing this paper. We also acknowledge M.-C. Arnaud, F. Béguin and the referees for their comments on the first version of this paper.
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Bonatti, C., Díaz, L. On maximal transitive sets of generic diffeomorphisms. Publ. math., Inst. Hautes Étud. Sci. 96, 171–197 (2003). https://doi.org/10.1007/s10240-003-0008-0
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DOI: https://doi.org/10.1007/s10240-003-0008-0