Why are there mini sets in the Mandelbrot Set?

One of the most interesting properties of the Mandelbrot Set is that we can find what appear to be miniature Julia Sets and mini Mandelbrot Sets.

It’s easier to look at a different question first: why is it that the Julia sets associated with a point on a Mandelbrot set tend to be similar to that region of the Mandelbrot set?

Consider the functions that create them:

Julia: \lim_{n\to\infty}(z\to z^2+k)^n(x)

Mandelbrot: \lim_{n\to\infty}(z\to z^2+x)^n(x)

And the question is: why is the region of the Mandelbrot set around a point x similar to a the Julia set where k=x?

It’s obvious that at that point, the two functions are the same, but, given that we’re studying chaos it seems odd that the slight variations in the region around it wouldn’t produce a totally different appearance.

There are a couple ways to approach this problem. The first is to note that this isn’t, at the vary least, necessarily true as demonstrated by the following experiment. I constructed a sequence of fractals formed by the function \lim_{n\to\infty} (z\to z^2 +0.3 +x/m)^n(x) where I varied m from one to a hundred along the integers:

(You may need to click to see the animation!)

(Images created with gnofract4d and animated with imagemaick.)

That only goes to dividing by a hundred, but the influence is pretty negligible at the end. I could divide by more until the influence was arbitrarily negligible.

The second way of looking at this is that Julia sets of similar k values tend to be similar and therefore the associated region of the Mandelbrot set is similar.

Finally, we can look at the fact that the Mandelbrot set is T_2 and thus, provided a point is not right on the edge of a sudden change, we can construct a set containing it and no the change (ie. select an arbitrarily small neighbourhood such that the change is negligible, as said previously).

The “mini-Julia sets” form when we get a miniature, often deformed, plane (let’s dub it a microplane, because making up terms is fun!). The microplane will have a root of the Mandelbrot set in its center.

If the microplane is small enough, it may be contained in one of the previously described neighbourhoods in which the iterated function is essentially a Julia function and thus form what appears to be a Julia set, though it is different in some respects (eg. connected, joined to the Mandelbrot, et cetera).

But the really odd things are the “mni Mandelbrot Sets”. What is going on with them?

I don’t really have a satisfactory answer. I do have a few observations, however:

Firstly, these `mini Mandelbrot Sets’ are quite different from the Mandelbrot Set. They seem to form on roots. They seem to have bulb-like Julia sets around them…

Here’s a visualisation of the formation of one:

It’s completely different from the formation of the Mandelbrot Set!

Finally, I’d like to put forward a theory: there is a natural propensity for an iterated function to form self-similarity.


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