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  "body": "\n## Definition\n\nLet $X$ be the [[Riemann sphere|0J78KBPFZB]] $\\hat{\\C}$ or the complex plane $\\C$, and let $f \\colon X \\to X$ be a [[holomorphic map|0NAHXR1FRH]].\nFor $n \\ge 0$, the [[iterate|AYVTR2ES5B]] $f^n$ is the $n$-fold composition $f \\circ \\cdots \\circ f$, with $f^0 = \\id$.\nThe pair $(X, f)$ is a *holomorphic* [[discrete-time dynamical system|5AA9AJEARJ]].\nThe *orbit* of a point $z \\in X$ is the sequence $z, f(z), f^2(z), \\ldots$.\n*Complex dynamics* is the study of the orbits of such systems and of how they depend on the point $z$ and on the map $f$.\n\n## Remarks\n\nThe standard cases are [[rational functions|0XDK9CGYKJ]] $f \\colon \\hat{\\C} \\to \\hat{\\C}$ of degree at least $2$, polynomials of degree at least $2$, and transcendental [[entire functions|0G4F0XB9X2]] $f \\colon \\C \\to \\C$.\nExample: $f(z) = z^2 + c$ with $c \\in \\C$ is a polynomial, hence holomorphic on $\\C$, so $(\\C, f)$ is a holomorphic dynamical system.\nNon-example: $f(z) = \\bar z^2$ is not holomorphic, since $\\partial f / \\partial \\bar z = 2 \\bar z \\ne 0$ for $z \\ne 0$, so its iteration is not part of complex dynamics in this sense.\nRational maps of degree $1$ are Möbius maps, whose dynamics is elementary, which is why the degree is usually taken to be at least $2$.\n",
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