Fatou’s pancake is not a widely-recognized term except in my own brain. Yet. But in honor of the auspicious occurrence of the math holiday I’ve dubbed Fatou’s Day, I would like to celebrate this set, the filled Julia set of the function f(z)=(z+z2)/2. It is vaguely pancake-shaped in honor of the other holiday today, Fat Tuesday, and it was investigated by Pierre Fatou himself in 1906. (You can read his article about it on the Bibliothèque Nationale de France website. It is in French, and the mathematical language he uses may be a bit old-fashioned.)
I’ve described the set I’m nicknaming Fatou’s pancake as both a bounded Fatou domain and a filled Julia set. What do those mean? There are several ways to define the sets. The one that makes the most sense intuitively is that you look at what happens to points in the complex plane when you plug them into a given function (so the Julia set is the Julia set for a particular function) and then plug the output into the function again and keep repeating that. A point can either stay bounded or run away to infinity. If it stays bounded, it’s in the filled Julia set, and if it runs away to infinity, it’s not. The Julia set is the boundary of the filled Julia set.
Fatou domains are a little less intuitive than the filled Julia set. But today is Fatou’s Day, darn it, so we’re going to jump into them! A Fatou domain is an open set that is invariant under the function f(z). That means every point in the domain is taken to another point in the domain. Additionally, the points in one Fatou domain behave similarly. (The exact meaning of “behave similarly” gets a bit technical. One way points could behave similarly is that they could all be attracted to the same point.) Taken together, the Fatou domains form the Fatou set, and the Julia set is everything left over. In my post last week about Fatou’s Day, I mentioned that you can think of the Fatou set as all the stuff that kind of does the same thing in clumps and the Julia set as the set of points around whom behavior is chaotic. If you’re in a Julia set, you don’t know whether a point near you will shoot off to infinity or stay politely bounded nearby.
An example is worth a thousand words, so let’s investigate the Julia and Fatou sets of the very well-behaved function f(z)=z2. Here, I’m using the variable z to remind us all that we’re living in the complex plane. If multiplication in the complex plane is unfamiliar to you, you might want to watch this beautiful video from Grant Sanderson, also known as 3 Blue 1 Brown.
The important things to understand for this post are that complex numbers can be thought of as vectors in the the plane with both a magnitude (distance from 0) and a direction (where the angle is measured counterclockwise from the x-axis) and that complex multiplication is not just stretching the way real multiplication is. (We can think about real multiplication as “stretching” or “squishing” the number line.) Complex multiplication has both a stretching and a rotating component. To multiply two complex numbers, we add their angles and multiply their magnitudes.
Thus, when we square a complex number, we double its angle and square its magnitude. If it’s really close to zero, with a magnitude less than 1, it will shrink in towards zero while rotating around. If it’s far from zero, with a magnitude greater than 1, it will get further from zero when we square it. If its magnitude is 1, the angle will double, so it will rotate counterclockwise by some amount, and the magnitude will stay the same.
The Fatou and Julia sets are based on the long-term behavior of points under repeated iteration of a function. So in the case of f(z)=z2, we don’t just square numbers once; we take an input, square it, square the square, and so on. To look at concrete numbers, we could start with the number i, or the square root of -1. It has a magnitude of 1 and an angle of 90 degrees, or π/2 if you want to be fancy and radian-y about it. Its trajectory would be i, -1, 1, and then it would stay there forever. On the other hand, the number 2 would just run along the x-axis to infinity: 2, 4, 8, 16, and so on. And a small number like 1/4+i/4 would spiral in towards 0: 1/4+1/4i, i/8, -1/64, 1/4096, and so on. In general, all points in the complex plane can be divided into three sets: points that spiral in towards 0, points that explode out towards infinity, and points that stay on the circle of radius 1. The latter points form the Julia set. The other two sets are the two Fatou components.
As Holly Krieger says in this fun video about filled Julia sets, the function f(z)=z2 is basically the only function with an easy-to-understand filled Julia set. The function that produces Fatou’s pancake, f(z)=(z+z2)/2, is considerably more difficult to analyze. If we choose a few numbers to plug in, we see that, for example, 0 stays put, as does 1. The number i eventually gets sucked in to 0. But numbers with large magnitudes end up running off to infinity. Eventually, the points that end up back at 0 are the bread of the pancake at the top of this post. Mathematically, that’s the filled Julia set or one of the Fatou domains. The boundary curve around the pancake is the Julia set. That curve is rather complicated, and rougher than it looks, though it has pleasant symmetry, both horizontal or vertical. It's not easy to get your hands on them with pen and paper, but you can explore Julia sets with online calculators. (Note: online Julia set calculators will generally work for functions of the form f(z)=z2+c, where you get to see what happens as you vary the variable c, rather than functions written in a form like f(z)=(z+z2)/2. The functions f(z)=z2+1/8 and f(z)=(z+z2)/2 have similar Julia sets.)
Whether you choose to make a filled Julia set out of flour and buttermilk or not, happy Fatou's Day!
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Möbius Strip
The Long Line
The Wallis Sieve
Two Tori Glued along a Slit
The Empty Set
The Menger Sponge
The Connected Sum of Four Hopf Links
The Sierpinski Triangle
Lexicographic Ordering on the Unit Square
The SNCF Metric
The Mandelbrot Set