Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know calculus but not much beyond. The Mandelbrot set is certainly fascinating in that you can zoom in and get an incredible amount of detail, all out of an analysis of the simple recursion $z\mapsto z^2+c$. So my plan is to show them a movie of a deep fractal zoom, and go over the definition of the Mandelbrot set. But I'd like to also show them something mathematically rigorous, and the main interesting properties I know about the Mandelbrot set are well beyond the scope of the course. I could mention connectedness, which is of course a seminal result, but that's probably not that interesting to someone at their level. So my question is whether anyone has any ideas about an interesting property of the Mandelbrot set that I could discuss at the calculus level, hopefully including an actual calculation or simple proof.

Solution 1:

I always think a discussion of the Mandelbrot set should go along with a discussion of the logistic family, as a simple (the simplest?) interesting model of population dynamics. Of course the logistic family is only the real part of the Mandelbrot set, and here is a first simple pre-calculus exercise: show how to change coordinates to get one parametrization from the other.

(Or: Even show that you can change coordinates for every complex quadratic polynomial to get one of the form $z^2+c$.)

The fact that the Mandelbrot set is bounded was already mentioned in a previous answer. It's easy and straightforward, and well worth covering.

Some other easy exercises would be to determine the fixed points of the polynomials, and the region where there is an attracting fixed point (derivative of modulus less than one), i.e. the "main cardioid" of the Mandelbrot set. Do the same thing for period 2, and maybe ask the question what happens with higher periods - see also comments about density of hyperbolicity below.

It is also possible to discuss the structure of the Julia set (phase space), and in particular the difference between disconnected Julia sets outside M and connected Julia sets inside. While a formal proof would be difficult at this level, giving the geometric idea is not too hard, and for very negative real c (i.e. large $\lambda$ in the logistic parametrization), the result that the invariant set is a Cantor set is easy to do with elementary means (this is done e.g. in Devaney's book "A first course on chaotic dynamical systems").

You say that connectivity of the Mandelbrot set might not be interesting to them, but it's always possible to tell the amusing story about how Mandelbrot's first computer pictures suggested that M is disconnected, but the editor of the paper carefully removed all the little islands from his picture, thinking they were dirt! You could accompany it by running two different algorithms (one which just does a pixel-by-pixel calculation and colors points white or black, making M look disconnected, and e.g. the more standard colored pictures that show the connectedness of level lines quite clearly), and thus making a point about the perils of computer experiments and the importance of mathematical proof (if you care about this).

Finally, I would say that it is a good idea to talk about density of hyperbolicity. The question whether every interior component of the Mandelbrot set corresponds to maps with an attracting cycle is one of the most important open questions in complex dynamics, and yet is quite easy to understand with just a little bit of experimentation. It is always good to show students that even seemingly innocent questions can be the subject of very difficult mathematical research. Of course this is also a chance to mention that density of hyperbolicity in the real case (i.e. density of period windows in the bifurcation diagram) was only established in the 90s, and was a major mathematical breakthrough.

I realize there is a lot here, and it may go beyond what you were looking for, so pick and choose!

Solution 2:

A proof that once |z|>2 then the recursion will take it to infinity.