An Example of Computing the Julia Set and Filled Julia Set

This example concerns the family of quadratic functions, Qc(z) = z^2 + c.

Question: What is the Filled Julia Set of Qc(z) when c = 0?

To answer this question, I must determine for what complex numbers z the orbit of z under Q0(z) is bounded.

We recall from basic mathematics that for real numbers x, the sequence x^k tends to 0 as k tends to infinity if |x| < 1. The sequence tends to infinity if |x| > 1.

Therefore, complex numbers in the interior of the unit circle are in the Filled Julia Set of Q0(z).

So the Julia Set of Q0(z) is the unit circle in the complex plane.