$C^r$ Stability Conjecture: structurally stable diffeomorphisms satisfy Axiom A

Let $M$ be a closed manifold: a compact Hausdorff smooth manifold without boundary, modelled on $\mathbb R^n$. Let $r\ge 1$ be an integer and let $f\in\mathrm{Diff}^r(M)$ be $C^r$ structurally stable: there is a neighbourhood $\mathcal U$ of $f$ in $\mathrm{Diff}^r(M)$ (with the $C^r$ topology) such that every $g\in\mathcal U$ is topologically conjugate to $f$. Then $f$ is hyperbolic in the sense of Axiom A: the nonwandering set $\Omega(f)$ is a hyperbolic set of $f$ and

This is the Palis–Smale $C^r$ stability conjecture as stated on the Open Problem Garden ("any $C^r$ structurally stable diffeomorphism is hyperbolic"). The case $r=1$ was proved by Mañé; for $r\ge 2$ it is open.

Formalization Note $r$ ranges over natural numbers $\ge 1$ (the case $r=\infty$ is not covered). "Hyperbolic" is read as Axiom A (definition StabilityConjecture_Basic); the strong transversality condition of the full Palis–Smale characterization is not part of the statement.