Smooth Ergodic Theory 1
$C^r$ Stability Conjecture
Structural stability is a global topological property: a dynamical system is structurally stable if all nearby systems have the same orbit structure, up to continuous reparametrisation. Hyperbolicity is a local differential property: the tangent bundle over the recurrent set splits into uniformly contracting and expanding directions. That these two conditions should be equivalent is one of the deepest principles in smooth dynamics. Conjecture ($C^r$ Stability Conjecture, Palis–Smale, ~1970) Let $M$ be a closed smooth manifold and $r \geq 1$. If $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable, then $f$ is hyperbolic, i.e., it satisfies Axiom A and the Strong Transversality Condition.