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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $N \\ge 2$.\nFor $i = 1, \\dots, N$ let $S_i \\colon \\R^n \\to \\R^n$ be a similarity with ratio $r_i \\in (0, 1)$, that is, $\\abs{S_i(x) - S_i(y)} = r_i \\abs{x - y}$ for all $x, y$.\nLet $p_1, \\dots, p_N \u003e 0$ with $p_1 + \\cdots + p_N = 1$.\nA Borel [[probability measure|059TJN0NZ1]] $\\mu$ on $\\R^n$ is *self-similar* for $(S_i, p_i)$ if\n$$\\mu = \\sum_{i=1}^{N} p_i \\, (S_i)_* \\mu,$$\nwhere $(S_i)_* \\mu = \\mu \\circ S_i^{-1}$ is the [[pushforward measure|3N66TM3TSQ]].\n\n## Remarks\n\nEach $S_i$ is a [[contraction|0QN69YA357]].\nHutchinson's theorem says that there is exactly one compactly supported such $\\mu$.\nIt is the fixed point of a contraction on a space of measures, and its support is the attractor of $(S_i)$.\nExample: on $\\R$ let $S_1(x) = x/2$, $S_2(x) = x/2 + 1/2$ and $p_1 = p_2 = 1/2$, and let $\\lambda$ be Lebesgue measure on $[0,1]$.\nThe measure $(S_1)_* \\lambda$ is uniform on $[0, 1/2]$ with density $2$, and $(S_2)_* \\lambda$ is uniform on $[1/2, 1]$ with density $2$.\nTheir average has density $1$ on $[0,1]$, so it equals $\\lambda$, and $\\lambda$ is self-similar.\nNon-example: with the same maps and $p_1 = 1/3$, $p_2 = 2/3$, the average has density $2/3$ on $[0, 1/2]$, so $\\lambda$ is not self-similar for these weights.\nThe Cantor measure is self-similar for $S_1(x) = x/3$, $S_2(x) = x/3 + 2/3$ and $p_1 = p_2 = 1/2$.\n",
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