The orbit is deterministic. The partial sums are Brownian. Compress hard enough and the skeleton reappears.
From Julia Sets to Brownian Motion
Iterate f_c(z) = z² + c backward: at each step, choose a random preimage branch z ↦ ±√(z - c). By Brolin’s theorem, the orbit converges to the equilibrium measure μ_c on the Julia set J(f_c). The Julia set is a repeller under forward iteration — forward orbits escape to attracting cycles — so backward iteration is the natural way to sample the fractal.
Given a Hölder observable φ (say, Re(z) or Im(z)), form the Birkhoff sums:
S_n = Σ (k=0 to n-1) [φ(z_k) - ∫ φ dμ_c]
The Almost-Sure Invariance Principle (Dupont 2010) guarantees that these deterministic partial sums can be coupled to a Brownian motion W with matched variance:
S_n = W(σ²n) + o(n^ε)
The error is sub-polynomial. For all polynomial-order statistics — variance, CLT, functional CLT — the Julia orbit is indistinguishable from true Brownian noise.
The Residual Question
If the coupling is so tight, what remains after you subtract the Brownian component? The ASIP says the error is o(n^ε) — small, but not zero. That residual carries the deterministic skeleton: the memory of the transfer operator’s spectral structure, the topology of the Julia set, the branch-selection correlations from backward iteration.
The strategy is compression. Apply a Haar wavelet decomposition to the path S_n, keeping only the coarse approximation. The residual R_n = S_n − Haar(S_n) isolates the fine-scale structure that the Brownian coupling cannot absorb.
Interactive Explorer
Explore backward-iterated Julia orbits across five parameter presets. Compare liminal paths against matched Brownian motion, then compress via Haar wavelets and examine what survives.
Liminal Motion
Julia Dynamics → Brownian via ASIP → Compress → Residualμc-sampled via backward iteration. ASIP: Sn = W(σ²n) + o(nε). E¹²=0.372 → Marcus.
What the Tabs Show
- Paths — Side-by-side comparison of Julia Birkhoff sums and Brownian motion with matched σ². At a glance, they are indistinguishable.
- σ² Conv. — Variance convergence: S²_n/n → σ²(c) for both processes. The single observable recovers only this scalar parameter.
- Increments — Histogram of raw increments. The Julia increments are not Gaussian (the measure μ_c is fractal), yet the ASIP couples their partial sums to a Gaussian process.
- Residual — Haar wavelet compression, residual autocorrelation, and ordinal pattern analysis. This is where the deterministic skeleton becomes visible.
- Levy Area — The quantity E¹² = Im(c)/2 determines whether the limiting SDE is Ito or Marcus. Nonzero imaginary part means nonzero Levy area.
- Theory — Summary of the mathematical framework.
The Only Distinguisher
Compress the path. Compute the residual. Measure its autocorrelation — the liminal residual shows persistent correlations decaying at the rate set by the transfer operator’s spectral gap. The Brownian residual is flat noise.
But the sharpest tool is ordinal patterns. For a sequence of increments, look at the relative ordering of consecutive windows of length D. A continuous deterministic map cannot produce all D! ordinal patterns — some are forbidden by the topology of f_c. Brownian increments, being i.i.d., produce all patterns with uniform frequency.
Forbidden ordinal patterns are the sole sub-exponential distinguisher between liminal motion and true Brownian noise.
Two parameters in the same hyperbolic component of the Mandelbrot set share identical forbidden sets. The forbidden patterns detect the topological dynamics — the monotone piece structure — but not the metric parameter c. Within a hyperbolic component, the deterministic skeleton is topologically universal.
Somewhere between the fractal and the Gaussian, the skeleton persists. Compress until only the structure remains.