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
→ EML Fractal Explorer — click to zoom, switch between all 8 operators, choose color schemes. Real-plane escape-time with live viewport.
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.