The filled Julia set

Section 3.2 The filled Julia set

When we iterate the function \(f(z)=z^2\) from any starting point \(z_0\in \mathbb C\) we find that there are two, very general possibilities:

The orbit stays bounded or
the orbit diverges to \(\infty\text{.}\)

In fact, this is true for any function \(f_c\) in the quadratic family, a fact that follows easily from theorem Theorem 3.2.1.

Theorem 3.2.1. The quadratic escape criterion.

Let

\begin{equation*} f_c(z) = z^2 + c. \end{equation*}

Then, the orbit of \(z_0\) diverges to \(\infty\) whenever \(|z_0|\) exceeds

\begin{equation*} R = \max(2,|c|). \end{equation*}

The number \(R\) is called the escape radius for the quadratic.

Proof.

Suppose that \(z_0\in\mathbb C\) satisfies \(|z_0| \gt 2\) and \(|z_0| \geq |c|\text{.}\) Then, by the reverse triangle inequality,

\begin{align*} |z_1| &= |f(z_0)| = |z_0^2 + c| \geq |z_0|^2 - |c|\\ & \geq |z_0|^2 - |z_0| = |z_0|\left(|z_0|-1\right) = \lambda |z_0|, \end{align*}

where \(\lambda \gt 1\text{.}\) Furthermore, \(z_1\) now also satisfies \(z_1 \gt R\) so that, by induction, \(|z_n| \geq \lambda^n |z_0|\text{.}\) As a result, \(z_n\to\infty\) as \(n\to\infty\text{.}\)

Note that this applies to any iterate \(z_i\) even if \(z_0\leq 2\text{.}\) As a result, once some iterate exceeds \(2\) in absolute value, the orbit is guaranteed to escape. As a result, a fundamental dichotomy arises whenever we iterate any function \(f_c\) in the quadratic family; this dichotomy gives rise the definition ofthe so-called filled Julia set of \(f_c\text{.}\)

Definition 3.2.2. The filled Julia set.

Given a point \(z_0\in\mathbb C\text{,}\) either:

The orbit of \(z_0\) stays bounded by \(R = \max(2,|c|)\) under iteration of \(f_c\text{,}\) in which case we say that \(z_0\) lies in the filled Julia set of \(f_c\) or
the orbit of \(z_0>\) diverges to \(\infty\) under iteration of \(f_c\text{,}\) in which \(z_0\) does not lie in the filled Julia set of \(f_c\text{.}\)