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.
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 \(g_{\lambda}(z) = \lambda z + z^2\text{.}\)
The escape radius
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.
Let \(\varphi(z) = z + (1-\lambda)\text{.}\) Show that
How is this observation related to the symmetry that we see in figure Figure 3.7.1?