Lecture 27

Julia sets

MCS 275 Spring 2021
Emily Dumas

Lecture 27: Julia sets

Course bulletins:

  • Project 3 due 6:00pm CDT on Friday March 19.
  • Some time today:
    • Project 3 autograder opens
    • Worksheet 10

Plan

Dynamics

Start with a function, e.g. $f(x) = x^2$, and a number, $a$.

Apply $f$ over and over again to get a sequence:

$a, f(a), f(f(a)), f(f(f(a))), ...$

This sequence $\{ f^{n}(a) \}_{n \geq 0}$ is the orbit of $a$ (under $f$). What can we say about it?

Let's start with a simple example, $f(x) = x^2$.

$2\to4\to16\to256 \to 65536\to\cdots$

$\frac{1}{2} \to \frac{1}{4} \to \frac{1}{16} \to \frac{1}{256} \to \frac{1}{65536} \to \cdots$

What happens for other starting points?

For real numbers, orbits under $f(x)=x^2$ are easy:

  • If $|a|<1$, the orbit of $a$ converges to $0$
  • If $|a|>1$, the orbit of $a$ goes to $\infty$
  • If $|a|=1$ the orbit is bounded but doesn't go to $0$

For $f(x)=x^2$ with complex numbers, it's the same!

  • If $|a|<1$, the orbit of $a$ converges to $0$
  • If $|a|>1$, the orbit of $a$ goes to $\infty$
  • If $|a|=1$ the orbit is bounded but doesn't go to $0$

$f(x)=x^2$

Filled Julia set

The filled Julia set of a polynomial $f(x)$, denoted $K_f$, is the set of complex numbers that have bounded orbits under $f$.

Named for mathematician Gaston Julia (1893-1978) who studied these sets starting in the 1920s.

$f(x)=x^2$

Next, let's look at $f(x) = x^2 - 1$.

First design

  • Write a function to test orbit behavior of a point.
  • Make a grid of points in $\mathbb{C}$ and test each one.
  • Record the results in an image file (black=bounded, white=unbounded).

Divergence criterion

For $f(x) = x^2-1$, if $|a|\geq 2$, then $f^n(a) \to \infty$ as $n \to \infty$.

Improved design

  • Make a grid of points in $\mathbb{C}$ and test each one.
  • Use numpy array operations to apply $f$ to all of them at once, repeatedly.
  • Optional: Keep track of ones that are already big, and don't apply $f$ to them.
  • Record the results in an image file (black=bounded, white=unbounded).

References

Revision history

  • 2021-03-17 Add second Julia notebook link
  • 2021-03-15 Add Julia notebook link
  • 2021-03-14 Initial publication