3 Adding an update for progress based on Alexandre Eremenko's suggestion.

Update: I have made some progress by following Alexandre Eremenko's suggestion to examine the orbits of the critical points. I would like to share an example application of this technique for identifying (rational) neutral cycles.

Consider the family of functions $f:z\to z - p(z)/p'(z)$, describing Newton-Raphson iteration on $p$, where $p$ is a cubic polynomial with simple roots. An element $f$ of this family has three attracting fixed points in $\mathbb{C}$, the roots of $p$, and is conjugate by a Möbius transformation to a function with an attracting fixed point at -1, an attracting fixed point at 1, and an attracting fixed point at $\lambda\in\mathbb{C}$. The following image is a visualization of types of cycles occurring in $f_\lambda$, for $\lambda$ in the axis aligned square of side length 0.04 with top left corner $-0.834 + 0.861 i$. The only candidate for a critical point of $f_\lambda$ is $\alpha = \lambda/3$.

In the image a positive green color component at a point $\lambda$ indicates that $f_\lambda^n(\alpha)$ converges to one of the fixed points (1, -1, or $\lambda$). Brighter green indicates slower convergence. The green regions are typical of Newton basin renderings. Points $\lambda$ having no green color component, but positive red and blue color components, indicate that $f_\lambda^n(\alpha)$ converges to a cycle of period greater than 1. In this case the cycle may be neutral or attracting. Brighter purple indicates slower convergence, so the bright purple regions indicate values of $\lambda$ such that $f_\lambda$ probably has neutral periodic points and $F(f_\lambda)$ might have parabolic components or Siegel disk components.

2 Changed "cycle of period $m$ at $z_0$" to "cycle of period $m$ at $w$" since tortoise and hare could detect a cycle starting at $z_0$ even if $z_0$ is merely pre-periodic.

Given a rational function $f$ on the Riemann sphere, I would like to answer the question: Does the Fatou set of the function, $F(f)$, contain any parabolic components or Siegel disk components? Additionally, if it turns out that $F(f)$ contains parabolic components or Siegel disk components I would like to visualize those components, perhaps by highlighting their boundaries in a plot of the Julia set of $f$. My question is:

Is there a known computational method for reliably detecting and describing the parabolic components and Siegel disk components of the Fatou set of an arbitrary rational function?

I am aware that a Siegel disk component will contain an indifferent periodic point of $f$ and that a parabolic component will have an indifferent periodic point of $f$ on its boundary, so a preliminary step in the method I seek might be to merely identify the indifferent periodic points of $f$. I have thought of a method for detecting these points, but I'm sure it is unreliable. The method is to take a large number of sample points, $z_0$, and use any cycle detection algorithm (say tortoise and hare) on the sequence $f^n(z_0)$. Then if there is that process detects a cycle of period $m$ at $w$ (equal to one of $z_0$, $f(z_0)$, $f^2(z_0)$, etc.), to numerically measure the magnitude of $(f^m)'(z_0)$ (f^m)'(w)$to see if that magnitude is tolerably close to 1. My secondary question is: Is there a known efficient and reliable computational method for identifying the indifferent periodic points of an arbitrary rational function? 1 # What are good methods for detecting parabolic components and Siegel disk components in the Fatou set of a rational function? Given a rational function$f$on the Riemann sphere, I would like to answer the question: Does the Fatou set of the function,$F(f)$, contain any parabolic components or Siegel disk components? Additionally, if it turns out that$F(f)$contains parabolic components or Siegel disk components I would like to visualize those components, perhaps by highlighting their boundaries in a plot of the Julia set of$f$. My question is: Is there a known computational method for reliably detecting and describing the parabolic components and Siegel disk components of the Fatou set of an arbitrary rational function? I am aware that a Siegel disk component will contain an indifferent periodic point of$f$and that a parabolic component will have an indifferent periodic point of$f$on its boundary, so a preliminary step in the method I seek might be to merely identify the indifferent periodic points of$f$. I have thought of a method for detecting these points, but I'm sure it is unreliable. The method is to take a large number of sample points,$z_0$, and use any cycle detection algorithm (say tortoise and hare) on the sequence$f^n(z_0)$. Then if there is a cycle of period$m$at$z_0$, to numerically measure the magnitude of$(f^m)'(z_0)\$ to see if that magnitude is tolerably close to 1. My secondary question is:

Is there a known efficient and reliable computational method for identifying the indifferent periodic points of an arbitrary rational function?