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 $$(f_a)_{a\in {\mathbb {R}}^k}$$ 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.