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Lanford's Proof of the Feigenbaum Conjectures

Speaker: Patrick Flynn
Date of Talk: April 28, 2025

1. The Motivating Example

Consider the functions \(f_ \mu : [0, 1]\to [0, 1]\) via \(x\mapsto \mu x (1 - x)\) for some positive real parameter \(\mu \). When performing fixed point iteration for sufficiently small \(\mu \), one gets exponential convergence to the stable nonzero fixed point of \(f_ \mu \); there’s nothing complicated. However, wher \(\mu\) is large (e.g. \(\mu = 4\)), the dynamics of this iteration scheme become chaotic, and sometimes these iterates do not converge at all. So somewhere in the middle, there ought to be a “phase transition” between these two drastically different behaviours.

This can be characterised by the derivative of \(f_ \mu \) at the nonzero fixed point \(x_*\). When \(f_ \mu ’ \left( x_ * \right) > -1\), standard arguments show exactly the exponential convergence. However, when \(f_ \mu ’ \left( x_* \right) < -1\), this stable fixed point bifurcates into a single unstable fixed point and two neighbouring stable \(2\)-periodic orbits. This is because \(f_ \mu \circ f_ \mu \) gets more fixed points when you increase \(\mu \).

Increasing \(\mu \) farther yet, one can expect the stable fixed points of \(f_ \mu \circ f_ \mu \) to bifurcate into \(4\)-periodic orbits, and those to bifurcate again, and so on and so forth.

2. Feigenbaum’s Conjecture

In the preceeding example, one can determine the bifurcation points \(\mu _0, \mu _1, \mu _2, \ldots\); naturally, one expects that these bifurcation points get closer and closer together. Feigenbaum’s first constant quantifies this, and one can find that \[\delta = \lim _{n\to\infty} \frac{\mu _n - \mu _{n-1}}{ \mu _{n+1} - \mu _n} \approx 4.669201609\ldots \] Perhaps what’s surprising is the fact that this is a generic phenomenon and applies to any family of functions with approximately the same shape as \(x(1-x)\).

Define the set \[M = \left\lbrace \psi \in C \left( \left[ -1, 1 \right] \right) : \psi (0) = 1, x \psi ’ (x) < 0\ \forall x\neq 0 \right\rbrace.\] Let \(\mathcal{D}\) be the subset of \(M\) for which, if \(a = -\psi (1)\) and \(b = \psi (a)\), one has \(b > a > 0\) and \(\psi (b) \leq a\). Define \(\mathcal{T} : \mathcal{D} \to M\) via \(\mathcal{T} \psi (x) = -\frac{1}{a} \psi \left( \psi (-ax) \right)\).

The motivation is that, given \(\psi \), one finds that \(I_0 = \left[ -a, a \right]\) and \(I_1 = \left[ b, 1 \right]\) are mapped into each other by \(\psi \). \(\mathcal{T}\) is called the doubling map, and it maintains a local maximum at \(x = 0\).

Conjecture 1. (Feigenbaum)

  1. \(\mathcal{T}\) has a fixed point \(g\).
  2. \(D_g\mathcal{T}\) has a simple eigenvalue \(\delta > 1\), and \(\sigma \left( D_ g \mathcal{T} \right) \setminus \left\lbrace \delta \right\rbrace \subseteq D_1(0)\).
  3. Let \(\Sigma _0\) be the bifurcation surface: \[\Sigma _0 = \left\lbrace \psi \in M : \exists x_0 \left( \psi ‘\left( x_0 \right) = -1,\ \left( \psi \circ \psi \right) ^{(3)}\left( x_0 \right) < 0 \right) \right\rbrace.\] Then this is intersected by the unstable subspace of \(D_ g \mathcal{T}\) transversely.

The statement of the second claim is somewhat loose; supposedly this takes place in some mysterious Banach space. Lanford does not prove the third claim in full generality, but rather gives a computer-assisted proof of the fact for the first example.

3. Computers and Fixed Points?

Lanford did not publish any pseudocode even, and “they did not have GitHub”, though code was published years later by Eckmann-Wittwer.

Unfortunately, Lanford’s computerised techniques were not expounded upon. It seems that with control over the spectrum of \(D_ g \mathcal{T}\), Banach’s fixed point theorem becomes helpful, and a computer can construct an approximate fixed point and somehow verify these fixed point conditions.

Magically, the fixed point \(g\) itself is analytic, despite giving the appearance of a discontinuous fractal-like function.