Finding a super-attractive lambda value

Suppose I want to find all the $\lambda$ values such that $g_{\lambda}(z)=\lambda z + z^2$ has a super-attractive orbit of period 3.

• What equation must I solve?
• Solve the equation (possible take home)
• Illustrate

Suppose I want to find all the $λ$ values such that $g_λ(z)=λz+z^2$ has a super-attractive orbit of period 3.

What equation must I solve?

$g_λ(g_λ(g_λ(z))))= ((λz+z^2+c)z+z^2+c)z+z^2+c$
This is my initial guess, but I don't think this is right. I was also thinking something along the lines of: $g_λ(g_λ(g_λ(z))))= λ(λ(λz+z^2+c)+(λz+z^2+c)^2+c)+(λ(λz+z^2+c)+(λz+z^2+c)^2+c)^2+c,$
but it got too confusing.

@Yousername Good try! First off, as we commented in class, I think this is a bit harder than I realized, because of the multiple slots for $z$. Thing that a super-attractive orbit of period two makes a lot more sense.

Having said that, I think that one critical issue is that you haven't written an equation - you've just defined a function. The point is that the critical point $-\lambda/2$ has to map back to itself. So your equation should be
$$g_{\lambda}(g_{\lambda}(g_{\lambda}(-\lambda/2)))) =-\lambda/2. $$

I believe this expands out to be:

$$\left(\left(z^2+\lambda z\right)^2+\lambda \left(z^2+\lambda z\right)\right)^2+\lambda \left(\left(z^2+\lambda z\right)^2+\lambda \left(z^2+\lambda z\right)\right)=-\frac{\lambda }{2}$$

But again - period 2 would be much more reasonable on a test!

Define $G_\lambda=g_\lambda^3$. Let $z_0$ be a critical point of $g(z)$. Thus, $z_0$ must satisfy $$g^\prime(z_0)=0$$ This has the solution $z_0=-\lambda/2$. Note that a critical point of $g_\lambda$ must also be a critical point of $G_\lambda$. Furthermore, an orbit under iteration of $g\lambda$ is super-attractive iff the orbit contains a critical point. Therefore, to find a super-attractive orbit of period $3$ under iteration of $g_\lambda$, we need to solve $$G_\lambda(z_0)=z_0$$ for $\lambda$. The values of $\lambda$ for which $z_0$ is both a critical point and a fixed point of $G_\lambda$ are $$\begin{align}\lambda=&-1.83187,\\ &-0.552675 - 0.959456 i,\\&-0.552675 + 0.959456 i,\\&0,\\&2,\\&2.55268 - 0.959456 i,\\&2.55268 + 0.959456 i,\\&3.83187\end{align}$$ These values are plotted on the following image.

mandelbrot_alternative_points.png718x370 23.1 KB