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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]].\nA [[periodic point|T75CZ5H47N]] $p$ of period $n$ is hyperbolic if no eigenvalue of $Df^n_p$ has modulus $1$.\nSuch a point has a stable manifold $W^s(p)$ and an unstable manifold $W^u(p)$.\nThe diffeomorphism $f$ is *Morse–Smale* if the following hold.\n\n1. The [[non-wandering set|EY2W34JZA4]] $\\Omega(f)$ is finite, so it consists of finitely many periodic orbits.\n2. Every periodic point of $f$ is hyperbolic.\n3. For all periodic points $p$ and $q$, the manifolds $W^s(p)$ and $W^u(q)$ intersect [[transversally|EWVATW6EYZ]].\n\n## Remarks\n\nFor a flow $\\varphi_t$ generated by a vector field on $M$, the non-wandering set must be the union of finitely many equilibria and closed orbits.\nAll of them must be hyperbolic, with transverse stable and unstable manifolds.\nExample: consider the flow on the round sphere $S^2$ that moves every point along its meridian from the north pole $N$ to the south pole $S$.\nLet $f$ be its time-one map.\nThen $\\Omega(f) = \\set{N, S}$, the point $N$ is a hyperbolic source and $S$ a hyperbolic sink.\nHere $W^s(S) = S^2 \\setminus \\set{N}$ is open, so it meets $W^u(N)$ transversally, and the other intersections are single points with full-dimensional manifolds or empty.\nNon-example: an irrational rotation of the circle has no periodic points, and its non-wandering set is the whole circle, so clause 1 fails.\nBy the stable manifold theorem for hyperbolic periodic points, $W^s(p)$ and $W^u(p)$ are injectively immersed submanifolds of $M$.\nSmale showed that the gradient flow of a Morse function satisfies clause 3 after a small change of the metric.\n",
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