Generic family with robustly infinitely many sinks

Abstract

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 $(f_a{)}_{a\in (-1,1{)}^k}$ 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 greater than $3$, the generic set is formed by families of diffeomorphisms. This result is a counter-example to a conjecture of Pugh and Shub.

to appear in Inventiones mathematicae

References in corpus (2)

• Abundance of non-uniformly hyperbolic Hénon like endomorphisms
• Normal forms and Misiurewicz renormalization for dissipative surface diffeomorphisms