Newton method and Siegel disks

Question

I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?

Answer 1

Implementing Alexandre's suggestion, I found that for $\lambda\approx 0.483096 + 1.00504\ i$, we have $$ \begin{align} P_{\lambda}(z) &= (z - 1)\ (z + 1/2 - \lambda)\ (z + 1/2 + \lambda) \\ &\approx \left(-1.02672+0.97106\ i\right)+\left(0.0267208-0.97106~ i\right)z+z^{3} \end{align} $$ is a good approximation to a polynomial whose corresponding Newton's method iteration function has a Siegel disk. While I found the parameters using Mathematica, you can use Sage to check them with this code.

Some dynamical pictures

The Newton's method basins of attraction looks like so:

If we zoom in on that component near zero, we see:

This certainly looks like a Siegel disk.

Finding the parameters

To find this thing, we first let $$ n_{\lambda}(z) = z-P_{\lambda}(z)/P_{\lambda}'(z) $$ and then $N_{\lambda}(z) = n_{\lambda}^3(z)$. We seek values of $z$ and $\lambda$ such that

$$\begin{align} N_{\lambda}(z) &= z \\ N_{\lambda}'(z) &= e^{2\pi i \left(\sqrt{5}-1\right)/2 }. \end{align}$$

There are other possible choices for the multiplier but I went with the golden angle.

Parameter pictures to help solve the system

To solve the system, we turn to Newton's method itself. More specifically, I used Mathematica's FindRoot command. To do so, we need initial guesses and we can use a picture to find those. First, the parameter space looks like so:

If we zoom in on the conspicuous looking blob at the top of the blue region, we see something like so:

The yellow dot at the center of the baby-brot is the super-attracting parameter $\lambda\approx 0.479 + 0.9997 \ i$ and satisfies $N_{\lambda}(0)=0$. It's not hard to find all such parameters since there's just a single equation of order 26 that needs to be solved.

If we zoom in a bit further, we see:

Now that second yellow dot on the cardioid oughtta be a good initial guess for the $\lambda$ we seek and a good initial guess for $z$ might be zero. Thus, using the initial guesses $\lambda\approx 0.4825 + 1.005 \ i$ and $z\approx 0.0$, I was able to find the more precise parameter stated at the beginning of this post.

Answer 2

All fixed fixed points of the Newton map $N(z)=z-P(z)/P'(z)$ are superattracting. So there are no invariant Siegel disks. But of course there are periodic Siegel disks of periods greater than $1$.

For example, for a one parametric family of polynomials of degree $3$ considered in Douady and Hubbard, On the dynamics of polynomial like maps, Ann sci. Ecole Norm. Super., 18(1985), in section VI,

there are values of parameter which correspond to an attracting cycle of period 3. These values form a cardioid-like region, a part of a "baby Mandelbrot set". On the boundary of this cardioid-like region, the multiplier of the 3-cycle takes all values of absolute value $1$. So you have a 3-cycle of Siegel disks for an appropriate value of this multiplier. (There are such multipliers that all holomorphic germs with this multiplier have a Siegel disk, for example, $\exp(\pi i\sqrt{2}).)$