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  "body": "\n## Definition\n\nThe *Feigenbaum–Cvitanović functional equation* is the [[functional equation|SRAN4EKWS0]]\n$$g(x) = -\\alpha \\, g\\big(g(x/\\alpha)\\big) \\quad \\text{for all } x \\in [-1, 1],$$\nin which the unknowns are a real number $\\alpha \u003e 1$ and a [[function|0EPAT51DXQ]] $g \\colon [-1, 1] \\to [-1, 1]$ such that:\n\n1. $g$ is even, that is, $g(-x) = g(x)$ for all $x \\in [-1, 1]$.\n2. $g(0) = 1$.\n3. $g$ is twice differentiable at $0$ with $g''(0) \u003c 0$, so that $0$ is a quadratic maximum.\n\n## Remarks\n\nSetting $x = 0$ in the equation gives $1 = -\\alpha \\, g(1)$, so any solution has $\\alpha = -1/g(1)$.\nLanford proved, with computer help, that there is a real-analytic solution, and its constant is the [[Feigenbaum constant|F1EHT2CPHC]] $\\alpha = 2.502907875\\ldots$.\nFor that solution $g(1) = -1/\\alpha \\approx -0.3995$.\n\nNon-example: a constant function $g = c$ does not solve the equation together with clause 2, since the equation gives $c = -\\alpha c$, so $c = 0$, while clause 2 asks $g(0) = 1$.\n\nThe right side is $T(g)$ for the renormalization, or period-doubling, operator $T(g)(x) = -\\alpha \\, g(g(x/\\alpha))$.\nSo the solution is a fixed point of $T$.\nThe other Feigenbaum constant $\\delta \\approx 4.6692$ is the only eigenvalue of modulus greater than $1$ of the derivative of $T$ at this fixed point.\n",
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