An archived instance of discourse for discussion in undergraduate Complex Dynamics.

Reasons for Features in 6.4


While the computational part of problem 6.4 in the Quadratic Family section is easy enough, I am not sure why the results relate to the features of the graph.

Take part a which states ''Show that the origin is an attractive (or super-attractive) fixed point of $g_\lambda$ whenever $|\lambda|<1$. This observation yields what prominent feature in figure 6.7?''

My computation is as follows

$$g_\lambda(z)=z^2+\lambda z\Rightarrow g_\lambda'(z)=2z+\lambda \Rightarrow g_\lambda'(0)=0+\lambda=\lambda.$$

So by deffinition, if $0<|\lambda|<1$ then the origin is an attractive fixed point.

However, I am not sure how this result relates to the graph. Is it simply the circle around the origin or is there more to it?



Your computation certainly looks good. As far as the interpretation goes, yes this is related to the circle centered at the origin. Specifically, your computation shows that the interior unit disk consists of all those $\lambda$ values such that the corresponding function $g_{\lambda}$ has an attracting (or super-attracting) fixed point.