Generic family with robustly infinitely many sinks

TL;DR

This paper demonstrates that for certain smooth families of maps on manifolds, there is a generic set where each map has infinitely many sinks, providing a counterexample to a longstanding conjecture.

Contribution

It constructs a generic family of smooth maps with infinitely many sinks, challenging the Pugh-Shub conjecture in dynamical systems.

Findings

Existence of a Baire generic set of families with infinitely many sinks

Counter-example to the Pugh-Shub conjecture

Applicable to manifolds of dimension ≥ 2

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.