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  "body": "\n## Definition\n\nLet $D$ be a subset of a [[metric space|0VKD1TP7F3]], for example of $\\C$, and let $f_1, f_2, \\dots$ be functions from $D$ to $D$.\nThe *forward* (or *inner*) compositions are $F_n = f_1 \\circ f_2 \\circ \\cdots \\circ f_n$.\nThe *backward* (or *outer*) compositions are $G_n = f_n \\circ f_{n-1} \\circ \\cdots \\circ f_1$.\nThe infinite forward composition converges at a point $z \\in D$ if the sequence $(F_n(z))$ has a [[limit|0XKWSSY8P2]], and similarly for the backward composition.\nWhen the limit exists for all $z$ in $D$, it defines the function $F(z) = \\lim_{n \\to \\infty} F_n(z)$ (or $G(z) = \\lim_{n \\to \\infty} G_n(z)$).\n\n## Remarks\n\nExample: on $D = \\R$ let $f_n(x) = \\frac{x}{2} + 1$ for all $n$.\nThen $G_n(0) = 2 - 2^{1-n}$, so the backward composition converges at $0$ to the fixed point $2$ of $f_1$.\nNon-example: with $f_n(x) = -x$ for all $n$, the sequence $F_n(1) = (-1)^n$ has no limit, so the infinite composition does not converge at $1$.\nFor a [[continued fraction|0BK6JBK58W]] with $f_n(w) = \\frac{a_n}{b_n + w}$, the values $F_n(0)$ are the convergents; backward compositions do not have this interpretation.\nWhen all $f_n$ equal a single map $f$, both $F_n$ and $G_n$ are the iterate $f^{\\circ n}$.\n\n## Explanations\n\n### L1 human\n\nForward compositions add each new function on the inside, next to the starting value.\nBackward compositions add it on the outside, so they describe a point moved step by step by a changing rule.\n",
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