Only squaring and absolute valueThe basics of complex dynamics are accessible with a mere few weeks of first-semester math. Or maybe you've seen complex numbers during high school. Complex numbers can be multiplied and have an absolute value. The only thing that's not intuitively clear is that "tends to infinity" for complex numbers means "grows unbounded in absolute value, regardless of any direction".
Let
f: ℂ → ℂ be a function. The
filled Julia set of
f consists of the points
z ∈ ℂ whose orbit under under
f remains bounded, i.e., the sequence
z,
f(
z),
f(
f(
z)),
f(
f(
f(
z))), ..., does
not grow to infinity. Show that the filled Julia set of
z ↦
z² is the closed unit disc { |
z| ≤ 1 }.
Show that the filled Julia set of
z ↦
z² − 2 contains no real number
x with |
x| > 2, and that it contains every real number
x with |
x| ≤ 2.
What happens if an orbit contains a point y with y > 2? You can then write y = 2 + a for some a > 0.
Harder though, probably needs more theory: Show that the filled Julia set of
z ↦
z² − 2 is precisely the real interval [−2, 2] ⊆ ℂ, i.e. there are no other numbers in it than we had already seen in the previous exercise.
Probably too hard to get right with only entry-level experience. I didn't get it proven properly on the fly. You'll have to consider arguments (angles) of the numbers as well as distance from 0, and if the argument under iteration becomes close to pure imaginary, you're fine, but it's not clear that this will always come close enough with enough distance from 0.
It's possible that this requires deeper theory about Julia sets, e.g., f(J) = J = inverse image of J under f.
The
Mandelbrot set M ⊆ ℂ is the set of points
c ∈ ℂ for which 0 is contained in the filled Julia set of
z ↦
z² +
c. We've already seen that 0 and −2 are in
M. Show that the intersection
M ∩ ℝ is the interval [−2, ¼].
The task has three parts. First, look at the case c < −2.
For the second case, −2 ≤ c ≤ 0, it may be helpful that |c²| < 2|c|. I'm still not getting this done on the fly, feel free to skip until I have reproven this.
For positive c, show that the orbit sequence 0, f(0), f(f(0)), ... is monotonically increasing, thus either converging to a limit point or growing unbounded. If a limit point x of the iteration under the continuous function f exists for real c, it must satisfy x = f(x) and must be a real number.
Show or refute that M is locally connected.When you receive your Fields medal for this, I would be grateful if you mentioned me for inspiration!
This is really all there is to the definition of the Mandelbrot set, and it shows why you can already draw nice pictures of these things in only 10 to 50 lines of code.
After that, you can get arbitrarily elaborate with coloring, depending on how fast values diverge to infinity. You may have seen the lovely smooth fractal zoom videos that abound on Youtube; they come from really elaborate programs that take hours and days to render the many frames.
-- Simon