## Section3.2The 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.

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 $f_c\text{.}$

###### Definition3.2.2The 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 $f_c\text{.}$