---
tag: 3C6P6X69P1
kind: definition
name: Feigenbaum–Cvitanović functional equation
msc: ["37E20", 39B12]
defines: Q7S0D1NAG6
uses: [0EPAT51DXQ, SRAN4EKWS0, F1EHT2CPHC]
sources:
  - url: https://en.wikipedia.org/wiki/Feigenbaum_constants
---

## Definition

The *Feigenbaum–Cvitanović functional equation* is the [[functional equation|SRAN4EKWS0]]
$$g(x) = -\alpha \, g\big(g(x/\alpha)\big) \quad \text{for all } x \in [-1, 1],$$
in which the unknowns are a real number $\alpha > 1$ and a [[function|0EPAT51DXQ]] $g \colon [-1, 1] \to [-1, 1]$ such that:

1. $g$ is even, that is, $g(-x) = g(x)$ for all $x \in [-1, 1]$.
2. $g(0) = 1$.
3. $g$ is twice differentiable at $0$ with $g''(0) < 0$, so that $0$ is a quadratic maximum.

## Remarks

Setting $x = 0$ in the equation gives $1 = -\alpha \, g(1)$, so any solution has $\alpha = -1/g(1)$.
Lanford proved, with computer help, that there is a real-analytic solution, and its constant is the [[Feigenbaum constant|F1EHT2CPHC]] $\alpha = 2.502907875\ldots$.
For that solution $g(1) = -1/\alpha \approx -0.3995$.

Non-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 $g(0) = 1$.

The right side is $T(g)$ for the renormalization, or period-doubling, operator $T(g)(x) = -\alpha \, g(g(x/\alpha))$.
So the solution is a fixed point of $T$.
The 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.