Exercise 2104Algebra › Complex numbersOlympiadAdded 3 Nov 2026

The Julia set of $z\mapsto z^2$

#Julia set#iteration#complex dynamics

Statement

Consider the iteration $z_{n+1}=z_n^2$ with $z_0\in\mathbb{C}$ . For which $z_0$ is the sequence $(z_n)$ bounded?

Hint

Write $z_0=re^{i\theta}$ and compute $z_n$ .

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Solution

Iteration. $z_n=z_0^{2^n}=r^{2^n}e^{i2^n\theta}$ .

Boundedness. $|z_n|=r^{2^n}$ .
- If $r<1$ : $r^{2^n}\to0$ , bounded.
- If $r=1$ : $|z_n|=1$ always, bounded.
- If $r>1$ : $r^{2^n}\to+\infty$ , unbounded.

Conclusion. The set of $z_0$ for which the sequence is bounded is the closed unit disk $\{z:|z|\le1\}$ . This is the filled Julia set of $z\mapsto z^2$ , and its boundary (the unit circle) is the Julia set.

The Julia set of $z\mapsto z^2$

Statement

Solution

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