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## Section3.7Exercises

###### 1

Show that the image of the circle of radius $|c|$ centered at $c$ under $f_c$ is the circle of radius $|c|^2$ centered at the origin.

In the next couple of problems, we'll try to get a grip on the family of functions $g_{\lambda}(z) = \lambda z + z^2\text{.}$

###### 2

The escape radius

1. Use the triangle inequality to show that $|g_{\lambda}(z_0) \geq |z_0|\text{,}$ whenever $|z_0| \gt 2\lambda\text{.}$ Conclude that the orbit of $z_0$ escapes whenever an iterate excedes $2\lambda\text{.}$
2. How does this compare to our general polynomial escape criterion?
###### 3

The escape locus for $g_{\lambda}$ (or any family of quadratics) is a partition of the complex parameter plane into two regions - one where the critical orbit stays bounded and one where the critical orbit diverges. This escape locus is shown in figure Figure 3.7.1. Let's try to understand a couple of things about this image.

1. Show that the origin is an attractive (or super-attracitve) fixed point of $g_{\lambda}\text{,}$ whenever $|\lambda| \lt 1\text{.}$ This observation yields what prominent feature in figure Figure 3.7.1?
2. Let $\varphi(z) = z + (1-\lambda)\text{.}$ Show that

\begin{equation*} g_{\lambda} \circ \varphi = \varphi \circ g_{2-\lambda}. \end{equation*}

How is this observation related to the symmetry that we see in figure Figure 3.7.1?