# Mandelbrot Set

1. May 22, 2006

### AKG

Someone asked me this on another forum: the main part of the Mandelbrot set is a big black blob with nonempty interior. If you look to the left of this blob (on the negative real axis) and zoom in, you will find more small blobs. Supposedly, these blobs are connected by a "thin filament". I guess what this means is that these thin filaments are subintervals of the negative real axis which are part of the Mandelbrot set, but being called "thin" suggests that for all points in this thin filaments, none of the points in the complex plane above or below these points are in the Mandelbrot set. (Does that make sense?) However, if you zoom in continually, do you not see blobs all the time? That is, if you see an interval of length L, say, connected two separate blobs, you can zoom in on this interval and find a tiny blob breaking up this interval of length L. So maybe you have two intervals, of length m and n with m + n < L. But if you zoom in on the interval of length m, you will find another tiny blob, right?

So is there in fact any part of the Mandelbrot set on the negative real axis that really is a line segment, or is it just a bunch of blobs, side by side?

For an analogy, consider the function $f : (0,1) \to \mathbb{R}$ defined by

$$f(x) = 1000000^{-\left \lceil \frac{1}{x}\right \rceil }$$

If you "zoom out" enough, then the region between the graph and the real-axis will just look like the line segment (0,1). If you zoom in a little, then you will see a rectangular region whose base is the interval [1/2, 1) on the real axis. Zoom in more, and two rectangles will become apparent, the one with base [1/2, 1) and the much smaller one with base [1/3, 1/2). At this point, all the rectangles to the left will be too small to notice, and so to the left of 1/3, it will still appear to be just a line segment. However, the more you zoom in, the more rectangles you will see, and we know that in fact, the region between the graph and the real-axis is all-rectangles, there are no thin intervals.

On the other hand, we could define f to be as it is above if $\lceil 1/x \rceil$ is odd, but define it to be 0 otherwise. Then the undergraph will be rectangles, connected by thin line segments, i.e. from right to left, it will alternate: rectangle, segment, rectangle, segment, etc.

So what is the Mandelbrot set like?