Palis conjecture: denseness of finitude of attractors

Let $M$ be a compact, connected, Hausdorff, boundaryless smooth manifold of dimension $n$ (modelled on $\mathbb R^n$), and let $r\ge1$ be an integer.

1. (Diffeomorphisms) There is a $C^r$-dense set $D\subset\mathrm{Diff}^r(M)$ such that every $f\in D$ has finitely many attractors $\Lambda_1,\dots,\Lambda_k$ whose basins cover Lebesgue-almost all of $M$:

2. (Flows) There is a $C^r$-dense set $D\subset\mathfrak X^r(M)$ of $C^r$ vector fields such that, for every $X\in D$, the flow $\Phi$ of $X$ has finitely many attractors whose basins cover Lebesgue-almost all of $M$.

Here an attractor is an invariant, transitive set whose basin $B(\Lambda)=\{p:\omega(p)\subset\Lambda\}$ has positive Lebesgue measure.

This is the conjecture of J. Palis as stated on the Open Problem Garden. It replaces the false hope that hyperbolic systems are dense by a statistical description of a dense set of systems.

Formalization Note Density is sequential $C^r$ density (IsCrDenseDiffeo, IsCrDenseVF). The flow part quantifies over every global flow of $X$; such a flow exists and is unique (see the flow-existence milestone).