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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $\\varphi \\colon \\R \\times X \\to X$, $(t, x) \\mapsto \\varphi_t(x)$, be a continuous [[flow|BBZP1R872N]].\nLet $\\mathcal{R}(\\varphi)$ be its [[chain recurrent set|6MX5R482KJ]].\nCall $x, y \\in \\mathcal{R}(\\varphi)$ chain equivalent if for all $\\varepsilon \u003e 0$ and $T \u003e 0$ there are $(\\varepsilon, T)$-chains from $x$ to $y$ and from $y$ to $x$.\nThe classes of this relation are the chain components.\nThen there is a continuous function $V \\colon X \\to \\R$, a complete [[Lyapunov function|01REYCJ371]] for $\\varphi$, such that:\n\n1. $V(\\varphi_t(x)) \u003c V(x)$ for every $x \\notin \\mathcal{R}(\\varphi)$ and every $t \u003e 0$.\n2. For $x, y \\in \\mathcal{R}(\\varphi)$, $V(x) = V(y)$ if and only if $x$ and $y$ lie in the same chain component.\n3. The set $V(\\mathcal{R}(\\varphi))$ is a compact [[nowhere dense|0VQYCCQVJ5]] subset of $\\R$.\n\n## Remarks\n\nThe proof uses attractor–repeller pairs $(A, A^\\ast)$.\nFor each such pair, Conley builds a continuous function equal to $0$ on $A$ and to $1$ on $A^\\ast$ that strictly decreases along orbits between them.\nA compact metric space carries only countably many attractors $A_1, A_2, \\ldots$, and $V = \\sum_n 3^{-n} V_n$ combines the corresponding functions.\nThe key fact is that $\\mathcal{R}(\\varphi)$ is the intersection of the sets $A \\cup A^\\ast$ over all attractor–repeller pairs.\nExample: let $f$ be a Morse function on a compact Riemannian manifold with pairwise distinct critical values.\nFor the flow of $-\\nabla f$, the chain recurrent set is the set of critical points, and $V = f$ satisfies the three conditions.\nCompactness is needed for the countability of attractors and for the existence of the attractor–repeller decomposition used in the proof.\n\n## Explanations\n\n### L1 human\n\nThe theorem splits every continuous flow on a compact metric space into two parts.\nOne part is the chain recurrent set, where the motion can come back to itself up to tiny errors.\nOn the rest, the motion runs steadily downhill for some height function, as water flows down a landscape.\nSo every flow on a compact space looks like a gradient flow once the recurrent pieces are collapsed to points.\n\n### L2 human\n\nThe function $V$ orders the chain components like levels of a landscape and shows that orbits outside $\\mathcal{R}(\\varphi)$ go from higher components to lower ones.\nThis is why the result is sometimes called the fundamental theorem of dynamical systems: the chain components are the indecomposable pieces, and between them there is only gradient-like motion.\nThe statement is the starting point for Morse decompositions and for the Conley index.\nVersions for continuous maps of compact metric spaces, with $\\varepsilon$-chains in place of $(\\varepsilon, T)$-chains, also hold.\n",
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