The shape of Julia sets

The boundary of the Mandelbrot set acts as a catalog for the shapes of Julia sets. That is, the Julia set corresponding to a point c in the boundary of the Mandelbrot set will have as part or most of its shape an infinite repetition of the shape of the Mandelbrot set near c. This has already been seen

• in image 3 on Page II where the cusp in the Mandelbrot set is reflected in the infinite number of cusps of the Julia set,
• in image 9 on Page III and image 12 on Page III where the tangencies between the main cardioid and the left and top "buds" of the Mandelbrot set are reflected in the repeated tangencies of the Julia sets, and
• in image 18 on Page IV where the branching near the Myrberg-Feigenbaum point is reflected in the infinite branching of the Julia set,

The next black and white image shows locations and small Julia set images of 8 points. The Julia sets are below the location images. Clicking on the black and white image gives a larger color image (85 kbytes) with somewhat enlarged (2 times) Julia sets and details of the points in the Mandelbrot set. The resemblance between the neighborhood of the point in the Mandelbrot set and the shape of the corresponding Julia set should be visible.

Each vertical column gives two points in one area of the Mandelbrot set. One point is the main branching point above a "bud," and the other point is the end of a major filament leaving that branch point. Every disk and cardioid has obvious sequences of buds on it that coverge to a point of tangency or a cusp. The buds we have chosen in the image below are (from left to right), (i) the top bud on the main cardioid; (ii) from the top bud, two buds to the right in the sequence to the cusp; (iii) from the top bud, two buds to the left in the sequence to the tangency at the left; (iv) the largest bud above the real axis on the disk to the left of the main cardioid.

The data for the four points along the top is:
c=-0.1010963638456138+0.9562865108091346, radius=0.175, skip=0, bands/color=1.
c=0.4379242413594628+0.3418920843381161, radius=0.075, skip=0, bands/color=2.
c=-0.670209187903254+0.4580609752969461i, radius=0.05, skip=0, bands/color=3.
c=-1.1623415998840382+0.2923689338965853i, radius=0.15, skip=0, bands/color=1.
25.
The data for the four points along the bottom is:
c=0+i, radius=0.175, skip=0, bands/color=1.
c=0.4445568792550437+0.4099331083009843i, radius=0.1, skip=0, bands/color=1.
c=-0.6356170288412094+0.4919204152038624i, radius=0.075, skip=0, bands/color=2.
c=-1.2963551381730362+0.4418516057351966i, radius=0.175, skip=0, bands/color=1.

The complications and twisting of the Julia sets increases as buds are chosen closer to the tangency of the main cardioid and the disk to the left, or closer to the cusp of the main cardioid. The points for the next Julia sets are located in a similar manner to the ones above but from buds much farther down the relevant sequence. We show one locator and two (almost identical looking) black and white images of Julia sets. Clicking on the locator gives a pair of images of the location neighborhoods. The left of the pair corresponds to the left of the two Julia sets.

Clicking on the black and white Julia sets gives larger, full color versions. These are prepared in a different manner from previous Julia sets, and have approximating bands in the same manner as the color images of the Mandelbrot sets. In the calculation of the Mandelbrot set, the value c is varied in f(z)=z*z+c and each band represents those values of c for which 0 takes a certain number of steps to get more than distance 2 from 0. In the calculation of the Julia sets, the value c is fixed and z is varied. Each band represents those values of z which take a certain number of steps to get more than distance 2 from 0. This will not give an accurate picture of the Julia set for points all points in the Mandelbrot set. It works for the Julia sets we show since there is only one basin of attraction and this consists of points that tend to infinity. We give image information for the Julia sets as well as for the Mandelbrot sets.

(The remaining color images obtainable from this page range in size from 32 kbytes to 144 kbytes.)

The first set of images are from the fifteenth largest bud in the main sequence of buds on the disk to the left of the main cardioid. For the branching point on the Mandelbrot set, the data is:
c=-0.7634732535193772+0.0906989952348076i, radius=0.00168, skip=35, bands/color=2.
For the end of the filament:
c=-0.7618389300387541+0.0959465324759709i, radius=0.00392, skip=14, bands/color=2.
26.
For both Julia sets, the color information is: skip=0, bands/color=2.

The next set of images are from a bud near the cusp on the main cardioid. For the branching point on the Mandelbrot set, the data is:
c=0.2713799273848036+0.0051344874807941i, radius=0.0025, skip=0, bands/color=2.
For the end of the filament:
c=0.2730013158136992+0.0058212644538372i, radius=0.0025, skip=0, bands/color=2.
27.
For both Julia sets, the color information is: skip=0, bands/color=2.

The first pair of Julia sets above are for c values that are near the tangency between the main cardioid and the disk to the left. Note the resemblance in gross outline between the Julia sets shown and the Julia set in image 9 on Page III. There is a similar resemblance in gross outline between the second pair and the Julia set in image 3 on Page II.

Superficial resemblances can be misleading. As shown in image 24 on Page V, there are resemblances between the branching point used in the top left item of image 25 above, and the branching point of the top bud on the secondary mandelbug whose cusp is at -1.75+0i. Compare the Julia set shown below with the top left Julia set in image 25. Clicking on the locator gives two images of the neighborhood of the relevant point. The left shows a large section of the neighborhood and the right shows the view in image 24 on Page V (with the same image information). The information for the left neighborhood is:
c=-1.7576944921030519+0.0176803441175033i, radius=0.2423, skip=0, bands/color=1.
28.
The image information for the Julia set is: skip=0, bands/color=1.

In case there seems to be no apparent resemblance between the Julia set shown in image 28 and the upper left Julia set of image 25, we present an enlarged image with extended coloring of the center of the Julia set in image 28. The center is at 0 and the rest of the image data is:
radius=0.2423, skip=-1, colors/band=3.
Recall that the radius of all the full Julia set images is 2.

Forward to The significance of the Mandelbrot set.
Back to What is the Mandelbrot set - Part V (6 embedded images).
Back to The Mandelbrot Set and Julia Sets.

Return to A Few Bits of Mathematics.
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matt at math.binghamton.edu