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  "body": "\n## Definition\n\nLet $p$ be a prime.\nLet $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$ extending the $p$-adic absolute value of $\\Q$.\nExamples are $\\Q_p$, its finite extensions, and $\\C_p$.\nA *$p$-adic dynamical system* is a pair $(X, \\varphi)$ in which:\n\n1. $X$ is either the projective line $\\mathbb{P}^1(K)$ or a subset of $K$, such as a closed disc;\n2. $\\varphi \\colon X \\to X$ is given by a rational function in $K(z)$ when $X = \\mathbb{P}^1(K)$, or by a power series with coefficients in $K$ that converges on $X$ otherwise.\n\n*$p$-adic dynamics* is the study of the [[iterates|AYVTR2ES5B]] $\\varphi^n$ and of the orbits $\\set{\\varphi^n(x) : n \\in \\N}$ of such systems, with respect to the topology of $\\abs{\\cdot}$.\n\n## Remarks\n\nExample: let $X = \\Z_p$ and $\\varphi(x) = x + 1$.\nIt maps $\\Z_p$ into $\\Z_p$, and it is a polynomial, so $(\\Z_p, \\varphi)$ is a $p$-adic dynamical system.\nIt is an isometry, since $\\abs{\\varphi(x) - \\varphi(y)} = \\abs{x - y}$.\nThe orbit of $0$ is $\\N$, which is dense in $\\Z_p$.\n\nNon-example: let $\\varphi(x) = x / p$ on $X = \\Z_p$.\nThen $\\varphi(1) = 1/p$ has absolute value $p \u003e 1$, so $\\varphi$ does not map $X$ into $X$, and clause 2 fails.\n\nA second example: $\\varphi(z) = z^2$ on the open unit disc $\\set{x \\in \\C_p : \\abs{x} \u003c 1}$.\nSince $\\abs{\\varphi^n(x)} = \\abs{x}^{2^n}$, every orbit tends to the attracting fixed point $0$.\n",
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