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  "body": "\n## Definition\n\nLet $\\overline{\\Q}$ be an algebraic closure of $\\Q$, let $N \\ge 1$ and let $h \\colon \\mathbb{P}^N(\\overline{\\Q}) \\to \\R$ be the absolute logarithmic [[height|F1FF505PWH]] on [[projective space|ZT9A6JK8QF]].\nLet $\\phi \\colon \\mathbb{P}^N \\to \\mathbb{P}^N$ be a morphism of degree $d \\ge 2$ defined over $\\overline{\\Q}$, and write $\\phi^n$ for its $n$-fold composition.\nThe *canonical height* of $\\phi$ is the function $\\hat{h}_\\phi \\colon \\mathbb{P}^N(\\overline{\\Q}) \\to \\R$ given by\n$$\\hat{h}_\\phi(P) = \\lim_{n \\to \\infty} \\frac{h(\\phi^n(P))}{d^n}.$$\n\n## Remarks\n\nThe limit exists because $h(\\phi(Q)) = d \\, h(Q) + O(1)$ uniformly in $Q$, so the sequence $d^{-n} h(\\phi^n(P))$ is Cauchy, by a telescoping argument.\nIt has these properties, due to Call and Silverman:\n\n1. $\\hat{h}_\\phi(\\phi(P)) = d \\, \\hat{h}_\\phi(P)$;\n2. $\\hat{h}_\\phi - h$ is bounded on $\\mathbb{P}^N(\\overline{\\Q})$;\n3. $\\hat{h}_\\phi(P) \\ge 0$, with equality exactly when $P$ is preperiodic for $\\phi$.\n\nProperty 3 uses Northcott's theorem: a point of canonical height $0$ has an orbit of bounded height inside a fixed [[number field|MWBDFS2KJ5]], and such an orbit is finite.\n\nExample: for $\\phi(z) = z^2$ on $\\mathbb{P}^1$, $h(\\phi(P)) = 2 h(P)$ exactly, so $\\hat{h}_\\phi = h$.\nIts zeros are $0$, $\\infty$ and the roots of unity, which are exactly the preperiodic points of $z \\mapsto z^2$.\n\nNon-example: for $\\psi(z) = z^2 + 1$ the naive height $h$ is not the canonical height of $\\psi$.\nIt fails property 1: $h(1) = 0$, but $h(\\psi(1)) = h(2) = \\log 2 \\ne 2 h(1)$.\n\nFor an elliptic curve $E$ over a number field, the [[Lattès map|9PK608W8MS]] that describes doubling on $x$-coordinates has degree $4$.\nIts canonical height, up to a normalizing factor, is the Néron–Tate height of $E$.\n",
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