Pugh–Shub conjecture on Kolmogorov-typical finiteness of attractors

Let $M$ be the manifold under consideration, and for $dgeq 1$ let a $d$-$C^r$-Kolmogorov typical property mean that, for every compact parameter manifold of dimension $d$, a generic $C^r$ family has the property for Lebesgue almost every parameter. A diffeomorphism displays finitely many sinks and other attractors when its attracting periodic or other attracting invariant sets are finite in number.

Pugh–Shub conjecture. For every $d\geq 1$, a $d$-Kolmogorov typical diffeomorphism displays finitely many sinks and other attractors.

This conjecture was proposed as a way to formulate typicality in parameter families without a natural Lebesgue measure on the full space of diffeomorphisms. It was disproved in the finite-regularity setting by the results described in the source, so the conjecture is refuted.