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  "body": "\n## Statement\n\nLet $M$ be a compact smooth manifold without boundary and let $f \\colon M \\to M$ be a $C^1$ diffeomorphism, with [[non-wandering set|EY2W34JZA4]] $\\Omega(f)$.\nSay that $f$ satisfies the strong transversality condition if, for all $x, y \\in \\Omega(f)$, the stable manifold $W^s(x)$ and the unstable manifold $W^u(y)$ intersect [[transversally|EWVATW6EYZ]].\nThen $f$ is [[structurally stable|KQX43WB57R]] in the $C^1$ topology if and only if $f$ satisfies [[Axiom A|S31RTTJPDC]] and the strong transversality condition.\n\n## Remarks\n\nThe statement was conjectured by Palis and Smale and is now a theorem.\nThe direction from Axiom A and strong transversality to structural stability was proved by Robbin for $C^2$ diffeomorphisms and by Robinson in the $C^1$ case.\nThe converse direction, that $C^1$ structural stability forces Axiom A and strong transversality, was proved by Mañé.\nIts proof uses closing-lemma techniques, in particular Mañé's ergodic version of the [[closing lemma|E1Z444F6TV]], and a delicate perturbation argument to obtain hyperbolicity from robustness.\nExample: hyperbolic toral automorphisms such as [[Arnold's cat map|BSCZN8Y1AN]] satisfy both conditions, so they are structurally stable.\nAn irrational rotation of the circle is not structurally stable, as nearby rational rotations have periodic orbits.\nAccordingly it does not satisfy Axiom A.\nThe analogous statement for flows was proved by Hayashi.\n\n## Explanations\n\n### L1 human\n\nA system is structurally stable if small changes of the rule do not change the pattern of its orbits.\nThe theorem says exactly which systems have this property.\nThey are the systems that stretch and squeeze uniformly on their recurrent part, and whose stable and unstable directions cross cleanly.\nSo robustness and hyperbolicity turn out to be the same thing.\n\n### L2 human\n\nThe theorem closes a programme started by Smale in the 1960s: to describe the structurally stable systems.\nThe easy half shows that hyperbolic systems are robust, using the persistence of hyperbolic sets under perturbation.\nThe hard half shows that robustness forces hyperbolicity; it relies on perturbation tools such as the [[closing lemma|E1Z444F6TV]].\nA related result is the $\\Omega$-stability theorem, which characterises the stability of the dynamics on $\\Omega(f)$ by Axiom A and the absence of cycles.\nIn the $C^r$ topology with $r \\ge 2$ the necessity direction is still open.\n",
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