# Are there an infinite number of minibrots on the real line?

This is at 25 zooms using Fractal Extreme. The red circle indicates that there are more minibrots inbetween the small one and the large one. The pattern of super big (bottom right), medium-sized (top-left), and two similar-sized mini ones inbetween seems to repeat as you zoom in and in. If you were to continue to zoom between the super big and small ones, would the pattern repeat indefinitely? Or would it eventually taper off like it does if you were to just go along the real line horizontally? Yes, there are there an infinite number of minibrots on the real line. Here is a direct demonstration of where an infinite number of such minibots may be found!

Consider the Misiurewicz point C=-1.54368901269207636, which is the real solution of the cubic equation $$C^3+2C^2+2C+2$$. Misiurewicz points have solutions which repeat, periodically, but do not go to zero. So, for the C value I gave, you can see the repeating pattern starting at x=0, and iterating $$x\mapsto x^2+C$$. Unlike hyperbolic points, which are attracting, Misiurewicz points are repelling and repeat. Hyperbolic centers of Mandelbrot bulbs also repeat, but they repeat by going to zero. In the neighborhood of a Misiurewicz point, the Mandelbrot is self similar as you zoom in. This also applies to the location of the mini-Mandelbrots in the Mandelbrot in the neighborhood of the Misiurewicz point. There is a mini-Mandelbrot at each of the cross hatches in this image. The cross hatch pattern repeats infinitely and is self similar as you zoom in. The baby mandelbrots at the cross hatch points get relatively smaller than the cross hatch pattern itself, but they're still there, repeating infinitely, as you zoom in infinitely. This supplies an infinite set of mini-Mandelbrots all on the real axis, and all in the neighborhood of this one particular Misiurewicz point. Of course, there are also an infinite number of other Misiurewicz points on the real axis, which have the same definition of being points where the pattern repeats without going to zero.

 0 (x_0, start at x=0)
-1.54368901269207636 (x_1=C; iterating x^2+C)
0.839286755214161133 (x_2=C^2+C)
-0.839286755214161133 (x_3=C^4 + 2C^3 + C^2 + C)
-0.839286755214161133 (x_4=X3^2+C ....)


This pattern repeats forever because C is the real valued solution of the algebraic equation $$x_3=-x_2$$. After factoring out roots for C=0, the algebraic equation reduces to the cubic equation I gave earlier, $$C^3+2C^2+2C+2=0$$.  • That is interesting. Unfortunately as I zoom in, the distance between the minibrots seem to grow, making it difficult to keep minibrots within the frame of the image without translating it.
– user80096
Aug 30, 2013 at 16:35
• Yes, the mini-Mandelbrots get smaller, relative to the cross hatches, as you zoom in. But if you have a window, then you can zoom in directly on the C value by any arbitrary amount, and then look at the biggest cross hatch on the left, and the biggest cross hatch on the right. These will look more or less the same, no matter how close you are to C. If you then chose to zoom in far enough on either of those cross hatches, there would be a mini-Mandelbrot at the center of the cross hatch. The chosen value of "C" guarantees the self similar property of the Mandelbrot as you zoom in. Aug 30, 2013 at 17:24

The Mandelbrot set is self-similar. So by zooming, you can get back the same picture. A nice animation of this can be found at this Wikipedia page (scroll down to "Examples").

So in your terms, the pattern would repeat infinitely.

• Depending on where you zoom in, it can also continually get denser as you zoom in. Aug 29, 2013 at 18:25
• @SheldonL That's the reason why I wrote "...you can get back..." :) Aug 29, 2013 at 18:38
• The Mandelbrot set is not exactly self-similar.
– lhf
Aug 30, 2013 at 17:52