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  "body": "\n## Definition\n\nFor $c \\in \\C$ let $f_c \\colon \\C \\to \\C$ be the polynomial $f_c(z) = z^2 + c$.\nThe family $(f_c)_{c \\in \\C}$ is the *quadratic family*.\n*Quadratic polynomial dynamics* is the [[complex dynamics|BQQZ0ZCGQF]] of this family.\nIt studies the [[iterates|AYVTR2ES5B]] $f_c^n$ for each $c$, and how they change with $c$.\nThe *filled Julia set* of $f_c$ is $K_c = \\set{z \\in \\C : (f_c^n(z))_{n \\ge 0} \\text{ is bounded}}$.\nIts boundary $\\partial K_c$ is the [[Julia set|5M4S7P04GR]] $J_c$ of $f_c$.\n\n## Remarks\n\nEvery [[polynomial|1G0B73WPX8]] $p(z) = a z^2 + b z + d$ with $a \\ne 0$ is conjugate to exactly one $f_c$ by an affine map $z \\mapsto \\alpha z + \\beta$.\nThis is why one parameter suffices.\nThe only critical point of $f_c$ in $\\C$ is $0$.\nThe set $K_c$ is connected if and only if the orbit of $0$ is bounded, and otherwise $K_c$ is a Cantor set.\nExample: for $c = 0$, $f_0^n(z) = z^{2^n}$ is bounded exactly when $\\abs{z} \\le 1$, so $K_0$ is the closed unit disc and $J_0$ is the unit circle.\nNon-example: $z \\mapsto z^3 + c$ is not in the quadratic family, since its degree is $3$.\n",
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