Work in progress. The interactive demo runs; the writeup below sketches the core ideas. A full treatment lives in the companion PDF.
The Problem of Noise
Every physical system that drifts, diffuses, or fluctuates is governed — at the density level — by a Fokker-Planck equation. The standard story writes particle dynamics as
dx(t) = -∇V(x) dt + σ dW(t)
where W is Brownian motion: structureless, memoryless, Gaussian. The density p(t,x) satisfies a PDE whose stochastic character enters solely through the driving signal.
If the randomness enters only through a signal, then the signal need not be random at all. It need only look random in the right statistical sense.
The Central Proposal
Replace the Brownian realisation with a signal generated by iterating f_c(z) = z^d + c on its Julia set. Three pillars support this:
Pillar 1 — Almost-Sure Invariance Principle. Birkhoff sums of Hölder observables along hyperbolic rational maps approximate Brownian motion with error O(n^(1/2 - ε)) (Dupont 2010). The orbit is deterministic; the statistics are Brownian.
Pillar 2 — Rough Path Homogenization. Chaotic fast-slow systems converge to SDEs in rough path topology, with a computable Lévy area correction (Kelly-Melbourne 2016-17). The rescaled process converges to an Itô diffusion with effective diffusion coefficient given by the Green-Kubo formula.
Pillar 3 — Orthogonal Polynomials on Julia Sets. The Bessis-Geronimo-Moussa construction provides a canonical orthogonal polynomial family on the equilibrium measure — a natural chaos expansion basis intrinsic to the deterministic system.
Live Demo
Three Julia sets provide six independent noise channels (Re/Im each), driving a three-body gravitational system. Switch tabs to see the different layers of the framework.
The Buddhabrot as Projection of Moduli
Each Julia set J(f_c) is a point in the moduli space of noise — the family parametrising deterministic noise models whose diffusion coefficients vary real-analytically within hyperbolic components.
The Buddhabrot arises as the fibre integral B(z) = ∫ O_c(z) dλ(c) — the pushforward of the orbit flow across moduli. Aggregating over c erases the moduli information.
The Buddhabrot shows what deterministic dynamics look like after moduli information is erased. Julia sets show that the erased information was geometrically real.
Information Hierarchy
- Brownian motion — σ² only.
- Fokker-Planck (σ²-matched) — Leading-order density evolution.
- Green-Kubo + autocorrelation — σ²(c) and decay rate λ(c).
- Spectral fingerprint (BGM) — Full Jacobi sequence encoding all moments of μ_c and fractal geometry.
- Moduli point c — Complete specification.
Known Obstacles
- The Lévy area is generically nonzero — the limiting SDE is Marcus-type for vector-valued observables.
- CLT failure at the Mandelbrot boundary — saving grace: Collet-Eckmann parameters carry full harmonic measure (Smirnov 2000).
- Singular measures kill exponential polynomial chaos convergence — capped at algebraic rates.
- Dimensional ceiling — one Julia set gives at most 2 real signals; need ⌈d/2⌉ sets.
- Computational overhead — potentially ~600x more efficient than 1000-trajectory Monte Carlo, but rough path scheme adds complexity.
Obstacles 1, 4, and 5 are essentially resolved for hyperbolic parameters. Obstacles 2 and 3 remain harder.
The orbit is deterministic. The statistics are Brownian. The structure is fractal. Somewhere in the liminal space between chaos and randomness, these limits meet.