Julia and Mandelbrot sets w or w/o Lyapunov sequences

[marcm200]

"Once around a cycle"

I was trying to color the interior of the Mset with path.length to see if there's a pattern. The path length is the summed up Euclidean distance of consecutive points in the attracting cycle, going once around.

Below are some examples for period 9 (A-C,E-F) and 8 (D). The distribution of the lengths is quite smooth, but other than that, it seems "random: Cardioids with the cusp being the cumulation point of short or long path lengths, or lob-sided (A-C), bulbs with short or long pathlengths towards the root point to the parent or lob-sided (D-F).

Is there a (hidden) regularity in that distribution within a component?

Technical details

• numerical calculations are done with double precision, 25000 max it. Cycles are determined by backtracing a full orbit.
• hyperbolic centers are used from my other thread about the root challenge (forum link)
• classification of a component as bulb or cardioid was done in an automated manner using the w-method (forum link).
• coloring was done (components are colored independently from one another) using a heatmap with the value \( \frac{L-L_{min}}{L_{max}-L_{min}} \) as a direct color index with L_min, L_max being the shortest and longest path length of a cycle within a given component, L being the path length of the current pixel.

[marcm200]

"Hull coloring"

To get interior structure into the Mset I colored a pixel here by a ratio of two areas: the area of the convex hull of the attracting periodic points and the area of the convex hull of the entire Julia set. This gives an interesting, island-like pattern (cosmic MJandelbrot background?).

• hull areas were estimated by pixel counting the outside in an image of size 2048 x 2048. The coloring is susceptible to this image size.
• the gray arc around the cardioid is a remnant of period-detection as I remove those pixels that do not have two layers of neighbours of the same period, leaving some further-out non-escaping, period-undetected  pixels.
• the image shows the period-5 minibrot at c=-1.256367930068180927-0.38032096347272276171*i, colored using an inverse heat-map.

[Adam Majewski]

does your color gradient has monotonic brightness ?
https://github.com/adammaj1/1D-RGB-color-gradient

[marcm200]

does your color gradient has monotonic brightness ?

I use the 7-color heatmap from http://www.andrewnoske.com/w/images/b/bf/Heatmap_gradient.png, with precomputed RGB values in a color table of usually only 800 entries. I guess the color transitions between index values could be smoothened, but I was satisfied here to see an inner structure in the Mandelbrot set.

[Adam Majewski]

I use the 7-color heatmap from http://www.andrewnoske.com/w/images/b/bf/Heatmap_gradient.png, with

try use monochrome heatmap. If the structures will be the same it is good, but when 7-color shows more structures then it means that these are artifacts cosuse by bad gradient ( not monotone)

[marcm200]

try use monochrome heatmap. If the structures will be the same it is good, but when 7-color shows more structures then it means that these are artifacts cosuse by bad gradient ( not monotone)

I use linear interpolation between the 7 fixed heatmap colors, so the red, blue and green channels are indepedently monotone, although not necessarily with the  same steepness or direction. Is this a concern for artefacts?

Below is a monochrome coloring: from black (RGB 0,0,0) to simple red (255,0,0) in a full color index (at least 256 entries). The structures prevail, I especially like the bright red to black sharp transition in the upper right.

[Adam Majewski]

Are the internal structures for other periods similar ?

I have not seen such structures before. What do you think about asking on the mathoverflow ?

[marcm200]

Are the internal structures for other periods similar ?

Below are a period 4 cardioid (c=-1.9407998065294851386) and a period 6 bulb (-1.7728929033816243077). The separated look of the latter comes from points where for some reason the convex hull calculation fails (too few points etc). So I left those c values blank. In general, the bulbs tend to be smoother, sometimes even showing no islands of sharp color change.

Longer cycles (period >= 7) are currently hard to compute with the initial implementation of a convex.hull algorithm (those would need larger Julia set images to get a decently high number of non-escaping points - or I have to switch to cell-mapping and use a covering).

I have not seen such structures before. What do you think about asking on the mathoverflow ?

Frankly, I have my doubts that the ratio of areas I use here for coloring bears a deeper mathemetical meaning. The convex hull was mainly chosen in the hope that by changing c slightly, the usually also slight change in the coordinates of the periodic points would sometimes lead to a periodic point now drifting outside its previous convex hull, thereby creating a new edge and in turn a different area value. But I'll keep that in mind for the future.

[pauldelbrot]

There is something mathematically meaningful going on here -- many of the blobs of color are Julia set shapes and they correspond to the position of the blob inside the bulb/cardioid.

[marcm200]

Here I tried a "digit coloring". For a pixel in the 2-bulb of the classic Mset, the attracting cycle was calculated and a value generated and formatted into scientific notation with unlimited zeros trailing. The digit in front of the radix is the first counted. Every digit 0-9 gets a distinct color.

Planet 2-bulb
1st image: 3rd digit of the imaginary part of the complex sum of the periodic points. For aesthetic reasons, the image was postprocessed: A sliding 16x16 window's average color was  finally set to its lower left corner pixel, then the image 1:4 downscaled.

Geodesics
2nd image, left: 3rd digit of real part of the sum. There appeared converging geodesics towards the child bulb's root points. Quite interesting!

Random number generator
2nd image, right: 6th digit of the path length of the 2-cycle. This seems to be getting quite random. If one were to construct a number from those pixels, let's say from top to bottom and from left to right: Will the digits be distributed in a random fashion? Is this number normal to base 10? Could it be used as a random number generator?