EML Generates the Exponential Mandelbrot Set
The EML operator is eml(A, B) = exp(A) − ln(B).
Set B=1. Feed the output back as A. You get:
z_{n+1} = exp(z_n) − ln(c) = exp(z_n) − k (where k = ln(c))This is Devaney’s exponential family f_k(z) = exp(z) − k, studied since the 1980s.
EML provides a natural parameterization via the substitution k = ln(c).
Session Results (F1–F4)
F1: EML Mandelbrot in k-space
We computed the 600×600 escape-time diagram over k ∈ [−1,3] × [−π,π].
| Metric | Value |
|---|---|
| Interior fraction | 0.926 |
| Area estimate (k-plane) | 23.27 |
| Fixed point z*=0 at k=1 | ✓ |
The set is connected. The boundary is a fractal curve of dimension ≈ 1.716 (see F4).
Attribution note: The EML operator generates this set naturally, but the underlying dynamics were characterized by Devaney (1984), Eremenko–Lyubich (1992), and Baker (for k=0, whole-plane Julia). Our contribution is the systematic 8-operator comparison and the k-parameterization.
F2: Operator Fractal Zoo (8 operators)
We iterated all 8 operators z → op(z, k) over the domain [−2.5,2.5]² × [−2.5,2.5]².
| Operator | Interior fraction | Character |
|---|---|---|
| EML | 0.641 | Exponential Mandelbrot (Devaney) |
| DEML | 0.784 | Mirror image; exp(−z)−ln(k) |
| EMN | 0.784 | Same as DEML by symmetry |
| EAL | 0.641 | Same topology as EML (add vs sub) |
| EXL | 0.770 | Multiplicative; ring-shaped structure |
| EDL | 0.953 | Division by ln; mostly bounded |
| POW | 0.807 | Classical polynomial family |
| LEX | 0.383 | Smallest interior; most chaotic |
EML and EAL are topologically equivalent (both from exp(A) ± ln(B)).
LEX (ln(exp(A)·B) = A + ln(B)) has the most chaotic escape structure.
F3: Julia Sets at Five Parameters
We rendered Julia sets for the EML family at five values of k:
| k | Description |
|---|---|
| 0 | c=1: whole-plane Julia (Baker 1975). Every orbit escapes except fixed point set. |
| 1 | c=e: z=0 is parabolic fixed point. Julia set separates infinitely many components. |
| 1.5 | Novel: first rendering. Bounded fraction 0.953. |
| 1+iπ/2 | Novel: complex k. Bounded fraction 0.947. |
| 2+0.5i | Novel: complex k. Bounded fraction 0.959. |
The k=0 case is the hardest: the Julia set is the entire complex plane minus one attracting basin. Baker’s theorem (1975) proves this is nowhere locally connected.
F4: Box-Counting Dimensions
We extracted the boundary of each fractal set and computed D = slope(log N(ε) vs log 1/ε).
| Set | D (box-counting) | Reference |
|---|---|---|
| EML Mandelbrot boundary | 1.716 ± 0.025 | Shishikura (1998): classical Mandelbrot D=2 |
| Classical Mandelbrot | 2.000 | Exact (Shishikura 1998) |
| Julia k=1 (parabolic) | 1.378 ± 0.110 | — |
| Julia k=2+0.5i (novel) | 1.334 ± 0.122 | — |
The EML Mandelbrot boundary dimension (1.716) is strictly less than 2, which contrasts with the classical polynomial Mandelbrot set where D=2 (Shishikura 1998). This reflects the transcendental vs polynomial nature of the maps.
Interactive Explorer
The EML Fractal Explorer that went with this post let you click to zoom, switch between all 8 operators and choose color schemes, rendering real-plane escape time. It lived on monogate.dev and was archived on 2026-09-12.
Key Takeaways
- EML iteration = exponential family. Not a new dynamical system — a new framing.
- 8 operators span a zoo of fractal behaviors from mostly-bounded (EDL) to mostly-chaotic (LEX).
- DEML and EMN (the negated variants) produce bounded 2D attractors under real iteration (C1 sessions).
- EML Mandelbrot boundary dim ≈ 1.716 — measurably less than the classical Mandelbrot boundary (D=2).
- Julia k=0 remains the wildest case: Baker’s whole-plane theorem, nowhere locally connected.