# Iteration Count vs Distance Estimate

# 1 Exterior

Calculate:

$$N = n + 2 - \log_2(\log(|z|^2))$$

$$d = \frac{\sqrt{|z|^2} \log(|z|^2)}{\left|\frac{dz}{dc}\right|}$$

with escape radius $|z|^2 > 256^2$.

Plot many points in blue on a log-log scale:

The green curve is approaching the tip of the antenna from the left, bounding like: if distance is smaller than d, iteration count must be larger than N(d). The relationship of the curve seems to be:

$$d = \frac{48}{4^N}$$

The red curve is approaching seahorse valley along an external ray, bounding like: if iteration count is larger than N, distance must be smaller than d(N). The relationship of the curve seems to be:

$$N = \frac{2.128\ldots}{\sqrt{d}}$$

Combining the bounds:

$$\frac{48}{4^N} < d < \frac{4.529\ldots}{N^2}$$ $$\log_4{\frac{48}{d}} < N < \frac{2.128\ldots}{\sqrt{d}}$$

Conjecture presented without proof.

# 2 Interior

The largest component of a given period $p = 2^k$ is probably in the period doubling cascade (iirc, need to check this…). It scales by Feigenbaum constant 4.6692… Feigenbaum constant (wikipedia). So $$d \approx a / 4.6692\ldots^{\log_2{p}} = a / p^{2.232\ldots}$$

The smallest component of a given period $p$ is nearest the tip of the antenna, with scaling by factor 16:

A new scaling along the spike of the Mandelbrot set, – Michael Frame, A.G. Davis Philip, and Adam Robucci, Computers & Graphics, Volume 16, Issue 2, 1992, Pages 223-234, ISSN 0097-8493, https://doi.org/10.1016/0097-8493(92)90050-6

Reprinted in

Chaos and Fractals: A Computer Graphical Journey (Chapter 40), ed. C.A. Pickover, 1998, pages 269-280, ISBN 978-0-444-50002-1 https://doi.org/10.1016/B978-044450002-1/50045-X

Child bulbs of top level cardioid scale approximately like $$\frac{1}{p^2} \sin\frac{q}{p},$$ conjectured for the $\lambda$ parameterization (which has top level discs, so no $\sin$ factor) of $M$ in

On the Dynamics of Iterated Maps III: The Individual Molecules of the M-Set, Self-Similiarity Properties, the Empirical $n^2$ Rule, and the $n^2$ Conjecture. – Benoit B. Mandelbrot, in Chaos, Fractals, and Dynamics, eds. P. Fischer and William R. Smith, 1985, ISBN 0-8247-7325-X https://archive.org/details/chaosfractalsdyn0000unse

A related conjecture is proven in

A Proof of the Mandelbrot $n^2$ Conjecture – John Guckenheimer and Richard McGehee, Report No, 15, 1984, Institut Mittag-Leffler https://pi.math.cornell.edu/~gucken/PDF/MandelbrotN2.pdf

and the size estimate is proven in

The size of Mandelbrot bulbs – A.C. Fowler and M.J. McGuinness, Chaos, Solitons & Fractals: X, Elsevier BV, 2019, vol 3, ISSN 2590-0544 https://doi.org/10.1016/j.csfx.2019.100019

These could provide bounds on interior distance vs period. Conjecture (probably could be sharpened):

$$d < \frac{2}{p^2}$$

Bounding period the other way is not possible because $d$ can get arbitrarily small in any component, but the size $s$ of the smallest components (assuming they are indeed the cardioid-like one nearest -2 for each $p$) seems to be

$$s \approx \frac{59.2\ldots}{16^p}$$

as $p \to \infty$, relative to the top level $p = 1$ cardioid with $s = 1$..

# 3 Applications

Efficient rendering of accurate images of the Mandelbrot set with ternary colouring:

• interior
• boundary / unknown
• exterior

without wasting time iterating further if the distance is surely going to be less than a pixel.

This sort of image can provide estimates of the area of the Mandelbrot set and the box-counting dimension of its boundary.

# 4 Caveats

Don’t forget the approximate nature of distance estimates!

For bounds on square grids, you need factors of 4 in both directions, and another factor of $\sqrt{2}$ for the diagonal vs edge of a square.

# 5 Numerical Verification

Output from a numerical search (wall-clock computation time 100mins using 16 threads on AMD 2700X CPU with 32GB RAM):

level

grid of 2^level cells to a side

total

4^level

int

guaranteed interior by distance estimate

bdry

might contain boundary

ext

guaranteed exterior by distance estimate

dim

estimate of box-counting dimension of boundary

ext00

counter-examples: large d, high N

ext01, ext10, ext11

other cases

int00

counter-examples: large d, high period

int01, int10, int11

other cases

maxn

potential counter-examples: high N, no d calculated as maximum iteration count was reached, no period detected

precision = 100 bits
c = 3.7489700317382812500000000000000e-01 + -2.0824050903320312500000000000000e-01 i
N = 3558118.22734
d = 2.53377e-24

precision = 53 bits
c = -0.170772552490234375 + -0.827236175537109375 i
N = 2504192738.88534
d = 2.1389e-112

precision = 53 bits
c = -1.052913665771484375 + -0.252559661865234375 i
N = 6432091876.80757
d = 3.274e-121

precision = 53 bits
c = -1.072231292724609375 + -0.239337921142578125 i
N = 2155547023.15851
d = 2.0166e-276