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  "body": "\n## Definition\n\nLet $M$ be a compact smooth manifold and let $f \\colon M \\to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].\nLet $\\Omega(f)$ be the [[non-wandering set|EY2W34JZA4]] of $f$.\nThen $f$ satisfies *Axiom A* if both of the following hold.\n\n1. $\\Omega(f)$ is a [[hyperbolic set|NY9MCZXY6N]] of $f$.\n2. The set of [[periodic points|T75CZ5H47N]] of $f$ is [[dense|0D198GPMTJ]] in $\\Omega(f)$.\n\n## Remarks\n\nEvery periodic point is non-wandering, so clause 2 says that $\\Omega(f)$ is the closure of the set of periodic points.\nExample: the north-south map of the circle, the time-one map of $\\theta' = \\sin\\theta$, has $\\Omega(f) = \\set{0, \\pi}$.\nBoth fixed points are hyperbolic, with derivatives $e$ and $e^{-1}$, so both clauses hold.\nNon-example: an irrational rotation of the circle has $\\Omega(f)$ equal to the whole circle, but it has no periodic points, and the derivative $1$ is not hyperbolic; both clauses fail.\nAnother non-example is the time-one map of $\\theta' = 1 - \\cos\\theta$: its only non-wandering point $0$ is fixed, but the derivative there is $1$, so clause 1 fails.\nFor flows, Axiom A asks that the non-wandering set be hyperbolic in the flow sense and that closed orbits be dense in it.\n",
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