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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[complete metric space|0BY52JW6NK]].\nAn *iterated function system* on $X$ is a finite family $f_1, \\ldots, f_N \\colon X \\to X$, with $N \\ge 1$, of [[contractions|0QN69YA357]]: there are constants $r_i \u003c 1$ with $d(f_i(x), f_i(y)) \\le r_i \\, d(x, y)$ for all $x, y \\in X$.\nIts *attractor* is a nonempty [[compact|00R0X1WGPX]] set $A \\subseteq X$ with\n$$A = f_1(A) \\cup \\cdots \\cup f_N(A).$$\n\n## Remarks\n\nHutchinson's theorem says that the attractor exists and is unique.\nThe proof applies the [[Banach fixed point theorem|0VK6VEH345]] to the map $K \\mapsto \\bigcup_i f_i(K)$ on the space of nonempty compact subsets of $X$ with the Hausdorff metric.\nThis space is complete, and the map is a contraction on it with constant $\\max_i r_i$.\nSo for every nonempty compact $K_0$, the iterates $K_{m+1} = \\bigcup_i f_i(K_m)$ converge to $A$.\n\nExample: on $\\R$, the maps $f_1(x) = x/2$ and $f_2(x) = x/2 + 1/2$ are contractions with constant $\\frac{1}{2}$, and $f_1([0, 1]) \\cup f_2([0, 1]) = [0, \\frac{1}{2}] \\cup [\\frac{1}{2}, 1] = [0, 1]$.\nSo the attractor is $[0, 1]$.\nWith $f_1(x) = x/3$ and $f_2(x) = x/3 + 2/3$ the attractor is the [[Cantor set|0BXNDZFQME]].\n\nNon-example: the family $g_1(x) = x/3$, $g_2(x) = x + 1$ on $\\R$ is not an iterated function system, since $g_2$ is not a contraction.\nIt has no nonempty compact set $A$ with $A = g_1(A) \\cup g_2(A)$: the maximum $a$ of such an $A$ would satisfy $a + 1 \\in A$, which is impossible.\n\nSuppose that the maps are similarities with ratios $r_i$ and that the pieces $f_i(A)$ do not overlap too much (the open set condition).\nThen the [[Hausdorff dimension|SSNFTHZ6YJ]] of $A$ is the number $s$ with $\\sum_i r_i^s = 1$.\n",
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