Inventiones mathematicae

, Volume 205, Issue 1, pp 121–172 | Cite as

Generic family with robustly infinitely many sinks

  • Pierre BergerEmail author


We show, for every \(r>d\ge 0\) or \(r=d\ge 2\), the existence of a Baire generic set of \(C^d\)-families of \(C^r\)-maps \((f_a)_{a\in {\mathbb {R}}^k}\) of a manifold M of dimension \(\ge \)2, so that for every a small the map \(f_a\) has infinitely many sinks. When the dimension of the manifold is \(\ge \)3, the generic set is formed by families of diffeomorphisms. When M is the annulus, this generic set is formed by local diffeomorphisms. This is a counter example to a conjecture of Pugh and Shub.



I thanks the referees for their advices and comments. I am very grateful to Enrique Pujals and Sylvain Crovisier for important conversations and their comments on the first version of this work. I am thankful to Jean-Christophe Yoccoz for his encouragements. This research was partially supported by the Balzan project of J. Palis, the French-Brazilian network and the project BRNUH of Université Sorbonne Paris Cité.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Université Paris 13VilletaneuseFrance

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