Recall the escape radius stated in the escape criterion for general polynomials - namely
\begin{equation} R = \max\left(2|a_n|,2\frac{|a_{n-1}|+\cdots+|a_0|}{|a_n|}\right). \tag{5.2.1} \end{equation}We can apply this directly to the quadratic family. It turns out, though, that we can do a bit better by focusing on this particular family.
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.
As with the polynomial escape criterion, 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. Note, though, that the quadratic escape criterion is a bit sharper than the version for general polynomials, when applied to the quadratic case. For example, theorem Theorem 1 implies that the escape radius for \(f_{-2}(z)=z^2-2\) is \(2\text{,}\) while equation (5.2.1) yields an escape radius of \(4\text{.}\) As we already know that the Julia set of \(f_{-2}\) is exactly the interval \([-2,2]\text{,}\) this estimate is as sharp as it can be. Furthermore, under the assumption that \(|c|\leq 2\text{,}\) the quadratic escape radius yields a uniform radius of \(2\text{.}\)