Feigenbaum–Cvitanović functional equation

Summary

The Feigenbaum–Cvitanović functional equation is $g(x) = -\alpha \, g(g(x/\alpha))$, for an even function $g$ with $g(0) = 1$ and a constant $\alpha$. Its solution is the fixed point of the renormalization operator that doubles the period of a unimodal map and rescales space. The constant $\alpha$ that comes with the solution is the second of the Feigenbaum constants, $\alpha \approx 2.5029$. The equation explains why the period-doubling cascade looks the same at every scale, for a large class of maps.