We show, for every r > d ≥ 0 or r = d ≥ 2, the existence of a Baire generic set of C d -families of C r -maps (f a ) a∈R k of a manifold M of dimension ≥ 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. 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.

In memoriam of Anosov's 80th anniversary. AbstractWe will introduce the notion of Emergence for a dynamical system, and we will conjecture the local typicality of super complex ones. Then, as part of this program, we will provide sufficient conditions for an open set of C d -families of C r -dynamics to contain a Baire generic set formed by families displaying infinitely many sinks at every parameter, for all ∞ ≥ r ≥ d ≥ 1 and d < ∞ and two different topologies on families. In particular the case d = r = 1 is new.