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  "body": "\n## Definition\n\nLet $K$ be an algebraically closed field that is complete for a nontrivial [[non-Archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$, such as [[$\\C_p$|FKF9YGBKHF]].\nThe *Berkovich affine line* $\\mathbb{A}^1_{\\mathrm{Berk}}$ is the set of maps $x \\colon K[T] \\to [0, \\infty)$, written $f \\mapsto [f]_x$, such that:\n\n1. $[f g]_x = [f]_x [g]_x$ for all $f, g \\in K[T]$;\n2. $[f + g]_x \\le \\max([f]_x, [g]_x)$ for all $f, g \\in K[T]$;\n3. $[c]_x = \\abs{c}$ for every constant $c \\in K$.\n\nIt carries the weakest topology for which $x \\mapsto [f]_x$ is continuous for every $f \\in K[T]$.\nThe *Berkovich projective line* is $\\mathbb{P}^1_{\\mathrm{Berk}} = \\mathbb{A}^1_{\\mathrm{Berk}} \\cup \\set{\\infty}$, the one-point compactification.\nEach point $a \\in K$ gives the point $[f]_a = \\abs{f(a)}$, so $\\mathbb{P}^1(K)$ is a dense subset of $\\mathbb{P}^1_{\\mathrm{Berk}}$.\nEvery [[rational map|0XDK9CGYKJ]] $\\varphi \\in K(z)$ of degree $d \\ge 1$ extends uniquely to a continuous map $\\varphi \\colon \\mathbb{P}^1_{\\mathrm{Berk}} \\to \\mathbb{P}^1_{\\mathrm{Berk}}$; for a polynomial $\\varphi$ it is given by $[f]_{\\varphi(x)} = [f \\circ \\varphi]_x$.\n*Berkovich space dynamics* is the study of the [[iterates|AYVTR2ES5B]] $\\varphi^n$ of this extended map on $\\mathbb{P}^1_{\\mathrm{Berk}}$, and more generally of endomorphisms of Berkovich analytic spaces.\n\n## Remarks\n\nExample: the *Gauss point* $\\zeta$ is given by $[\\sum_i c_i T^i]_\\zeta = \\max_i \\abs{c_i}$.\nIt is multiplicative by Gauss's lemma, satisfies the strong triangle inequality, and restricts to $\\abs{\\cdot}$ on constants, so it lies in $\\mathbb{A}^1_{\\mathrm{Berk}}$.\nIt is the supremum norm on the closed unit disc, and it is not of the form $[f]_a$ for any $a \\in K$.\nFor $\\varphi(z) = z^2$, the series $f(T^2)$ has the same coefficients as $f(T)$, so $[f]_{\\varphi(\\zeta)} = [f]_\\zeta$ and $\\zeta$ is a fixed point.\n\nNon-example: $[f] = \\max(\\abs{f(0)}, \\abs{f(1)})$ satisfies clauses 2 and 3 but not clause 1.\nIndeed $[T] = 1$ and $[T - 1] = 1$, but $[T(T - 1)] = 0$.\n\nBerkovich showed that $\\mathbb{P}^1_{\\mathrm{Berk}}$ is compact, Hausdorff and path-connected.\nThe Berkovich line is the analytification of $\\mathbb{P}^1$; a related but different notion of analytic space is the [[rigid analytic space|YAH6MZVG7K]].\n",
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