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  "body": "\n## Description\n\nFor $c \\in \\C$ let $f_c(z) = z^2 + c$, the [[quadratic family|WDZFV2ACXR]].\nThe *Mandelbrot set* is\n$$M = \\set{c \\in \\C : \\text{the sequence } f_c^n(0),\\ n \\ge 0, \\text{ is bounded}}.$$\nHere $0$ is the only critical point of $f_c$.\nCheck of a few points:\n\n1. $c = 0$: the orbit of $0$ is $0, 0, 0, \\ldots$, so $0 \\in M$.\n2. $c = -1$: the orbit is $0, -1, 0, -1, \\ldots$, so $-1 \\in M$.\n3. $c = i$: the orbit is $0, i, -1 + i, -i, -1 + i, -i, \\ldots$, so $i \\in M$.\n4. $c = 1$: the orbit is $0, 1, 2, 5, 26, \\ldots$, which is unbounded, so $1 \\notin M$.\n\nIf some $\\abs{f_c^n(0)} \u003e 2$, the orbit tends to $\\infty$.\nSo $M$ is the set of $c$ with $\\abs{f_c^n(0)} \\le 2$ for all $n$, and $M$ is a compact subset of the disc $\\abs{c} \\le 2$.\n\n## Remarks\n\nA parameter $c$ lies in $M$ exactly when the [[Julia set|5M4S7P04GR]] of $f_c$ is connected.\nFor $c \\notin M$ the Julia set is a Cantor set.\nDouady and Hubbard proved that $M$ is connected.\nIts intersection with the real line is the interval $[-2, 1/4]$.\nShishikura proved that the boundary of $M$ has [[Hausdorff dimension|SSNFTHZ6YJ]] $2$.\nThe main cardioid is the set of $c = \\lambda/2 - \\lambda^2/4$ with $\\abs{\\lambda} \u003c 1$.\nFor these $c$, the map $f_c$ has a fixed point with multiplier $\\lambda$, hence an attracting fixed point.\nWhether $M$ is [[locally connected|FDVAHRJKF3]] is an open problem, known as the MLC conjecture.\n\n## Explanations\n\n### L1 human\n\nFor each complex number $c$, start at $0$ and apply $z \\mapsto z^2 + c$ again and again.\nColour $c$ black if the numbers you get stay bounded, and leave it white if they run off to infinity.\nThe black region is the Mandelbrot set, a cardioid with discs and filaments attached, and with smaller copies of itself everywhere along its edge.\nIt is a map of the whole quadratic family: each point tells whether the Julia set of that map is one connected piece or dust.\n",
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