The purpose of this blog post is to give a short, constructive, computation-free proof of the following theorem:
Theorem: Every compact subset of the Riemann sphere can be arbitrarily closely approximated (in the Hausdorff metric) by the Julia set of a rational map.
A rational map is just a ratio of (complex) polynomials. Every holomorphic map from the Riemann sphere to itself is of this form. The Julia set of a rational map is the closure of the set of repelling periodic points; it is both forward and backward invariant. The complement of the Julia set is called the Fatou set.
Kathryn Lindsey gave a nice constructive proof that any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Her proof depends on an interpolation result by Curtiss. Kathryn is a postdoc at Chicago, and talked about her proof in our dynamics seminar a few weeks ago. It was a nice proof, and a nice talk, but I wondered if there was an elementary argument that one could see without doing any computation, and today I came up with the following.
The construction depends on the idea of an electromagnetic dipole. This is a pair of charged particles of equal but opposite charge; from far away, the two charges almost cancel, and the particle-pair is effectively neutral. The analog of a dipole for a rational map is a factor of the form where is small; this is a function which is uniformly close to 1 far from the pair , . I like the idea of a dipole, as a kind of “component” of a rational function which modifies it in a localized, predictable way, and wonder if it has further uses.
If is any rational function, and is a point in the Fatou set of , we can build a new function by multiplying with a dipole centered at whose zero-pole pair are apart. As , we claim (technical assumption: assuming has no indifferent fixed points or Herman rings) that the Julia sets of converge in the Hausdorff topology to a set , which consists of the union of the Julia set of , together with the point and its preimages under . This is easy to see: on the complement of the set , the dynamics of converges uniformly to that of . On the other hand, maps a small neighborhood of over the complement of at most one point in the Riemann sphere; thus this neighborhood contains some point in the Julia set, and likewise for every point in the preimage of . This proves the claim.
Now suppose we want to approximate the set (for convenience, suppose is disjoint from the unit circle). We can start with where is very large, choose a finite set which approximates in the Hausdorff sense ( is a “pixelated” version of – hence “Pixie dust”), and multiply by a dipole centered at each point of . If is very big, the Julia set is as close as we like to the union of with the unit circle. But after conjugating by a conformal map, the unit circle can be made as small as we like, and moved near any point we like, say some point of . This completes the proof.
It is interesting that although such rational functions are essentially trivial to write down, drawing their Julia sets is bound to be disappointing. This is because when the zero-pole pair of the dipole are very close, the dipole is numerically indistinguishable from the constant function 1 at the resolution of the pixels in a drawing.
Here are four examples, with , and 80 dipoles, with . The dipoles spell out a faint pixelated “HI” at the top of each figure, and the prominent circle is (close to) the unit circle.
When I mentioned this construction to Curt McMullen, he alerted me to another preprint by Oleg Ivrii, which gives another, quite different, construction of a polynomial with quasi-circle Julia set which approximates any given Jordan curve (apologies if there are alternate constructions by other people that I have not mentioned).
(Update November 4:) Oleg Ivrii gives yet another (even shorter!) construction of a Julia set approximating any closed set, in the comments below.
(Update November 13:) Merry Xmas!