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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nFor $\\varepsilon \u003e 0$ and $x, y \\in X$, an *$\\varepsilon$-chain* from $x$ to $y$ is a finite sequence $x = x_0, x_1, \\ldots, x_n = y$ with $n \\ge 1$ and $d(f(x_{i-1}), x_i) \u003c \\varepsilon$ for $i = 1, \\ldots, n$.\nA point $x \\in X$ is *chain recurrent* if for every $\\varepsilon \u003e 0$ there is an $\\varepsilon$-chain from $x$ to $x$.\nThe *chain recurrent set* $\\mathcal{R}(f)$ is the set of chain recurrent points.\n\n## Remarks\n\nFor a continuous flow $\\varphi_t$, an $(\\varepsilon, T)$-chain from $x$ to $y$ is a sequence $x = x_0, \\ldots, x_n = y$ with times $t_1, \\ldots, t_n \\ge T$.\nIt must satisfy $d(\\varphi_{t_i}(x_{i-1}), x_i) \u003c \\varepsilon$ for each $i$.\nA point $x$ is chain recurrent for the flow if such chains from $x$ to $x$ exist for all $\\varepsilon \u003e 0$ and $T \u003e 0$.\nThe set $\\mathcal{R}(f)$ is closed and contains the [[non-wandering set|EY2W34JZA4]] $\\Omega(f)$.\nExample: for the identity map, $x_0 = x_1 = x$ is an $\\varepsilon$-chain from $x$ to $x$ for every $\\varepsilon$, so every point is chain recurrent.\nNon-example: let $f(x) = x/2$ on $[0, 1]$ and $x \u003e 0$, $0 \u003c \\varepsilon \u003c x/2$.\nIf $x_{i-1} \\le x$, then $x_i \u003c x_{i-1}/2 + \\varepsilon \u003c x$, so by induction every $x_i$ with $i \\ge 1$ is less than $x$; no $\\varepsilon$-chain returns to $x$.\nFor this map $\\mathcal{R}(f) = \\set{0}$.\n\n## Explanations\n\n### L1 human\n\nAn $\\varepsilon$-chain is an orbit with small errors allowed at each step.\nA point is chain recurrent if it can come back to itself along such chains, for every size of error.\n",
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