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  "body": "\n## Example\n\nLet $\\Lambda \\subset \\C$ be a [[lattice|WN9V7WSJKF]], let $T = \\C / \\Lambda$ be the [[complex torus|Y7MA6KBE9M]], and let $\\wp$ be the Weierstrass [[elliptic function|FF9ZPCZ59K]] of $\\Lambda$, with invariants $g_2, g_3$.\nThe duplication formula expresses $\\wp(2u)$ as a [[rational function|0XDK9CGYKJ]] of $\\wp(u)$:\n$$\\wp(2u) = L(\\wp(u)), \\qquad L(x) = \\frac{x^4 + \\frac{g_2}{2} x^2 + 2 g_3 x + \\frac{g_2^2}{16}}{4x^3 - g_2 x - g_3}.$$\nThe map $L$ of degree $4$ is a *Lattès map*.\nIt is the shadow of the doubling map $u \\mapsto 2u$ on $T$ under the projection $\\wp \\colon T \\to \\hat{\\C}$.\n\nFor a lattice with $g_2 = 4$ and $g_3 = 0$, which exists because $g_2^3 - 27 g_3^2 = 64 \\ne 0$, the formula gives\n$$L(x) = \\frac{(x^2 + 1)^2}{4x(x^2 - 1)}.$$\n\nIts [[Julia set|5M4S7P04GR]] is the whole Riemann sphere.\nThe reason: the doubling map on $T$ multiplies small distances by $2$.\nSo the image under $u \\mapsto 2^n u$ of any small disc in $T$ covers $T$ for $n$ large.\nApplying $\\wp$, for every nonempty open set $U \\subseteq \\hat{\\C}$ some iterate $L^n$ maps $U$ onto $\\hat{\\C}$.\nThe Fatou set is invariant and the Julia set is nonempty, so an open subset of the Fatou set never maps onto $\\hat{\\C}$.\nHence the Fatou set of $L$ is empty.\n\n## Remarks\n\nLattès maps are the standard examples of rational maps whose Julia set is the whole sphere.\nIn general, a Lattès map is a rational map induced by an affine self-map $u \\mapsto a u + b$ of a complex torus, with $\\abs{a}^2 \\ge 2$ an integer.\nThe map passes to the sphere through the quotient of the torus by a finite group of automorphisms.\n\n## Explanations\n\n### L1 human\n\nOn a doughnut-shaped torus, doubling every point stretches each small patch until it wraps around the whole torus.\nThe Weierstrass function folds the torus onto the sphere, and the doubling becomes a rational map of the sphere.\nEvery small region of the sphere gets spread over the whole sphere after a few steps, so the map is chaotic everywhere.\nThere is no calm region at all: the Julia set is the entire sphere.\n",
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