What are good methods for detecting parabolic components and Siegel disk components in the Fatou set of a rational function?

It depends on how your function is given, and what exactly you mean by "detecting". Anyway, you don't have to look at the "large number of sample points $z_0$.

It is enough to look at the critical points. In principle, all neutral rational cycles and Siegel discs can be found from the behavior of the orbits of critical points.

Every attracting cycle and every rational neutral cycle attracts some critical point (Fatou's theorem). Find many points of the orbit of every critical point. If you see an orbit converging to a cycle, you can easily tell an attracting cycle from a neutral rational one: convergence to an attracting cycle is geometric, while convergence to a neutral rational cycle is EXTREMELLY slow.

Detecting a Siegel disc, it is much harder. It is known that the orbits of critical points are dense on the boundary of a Siegel disc. But this is of little help.

But at least a picture of the orbits of critical points can give you the idea of what the preiod of your Siegel disc could possibly be. Then you just find the fixed points of the corresponding iterate and see whether any of them is neutral.

Actually, if your function is given numerically, it is unlikely that it has a neutral point, so there is nothing to detect...

For detecting Siegel disc around fixed point in a plot of the Julia set of f one can use average discrete velocity of orbit.

(source)

It is described here

https://commons.wikimedia.org/wiki/File%3AGolden_Mean_Quadratic_Siegel_Disc_Speed.png

I have learned this method from Chris King

http://www.dhushara.com/DarkHeart/DarkHeart.htm

To highlight boundaries of Siegel disk components in a plot of the Julia set of $f$ one can use this method :

(source)

It works for cases when critical point is on the boundary of Siegel disc component.

See : Building blocks for Quadratic Julia sets by : Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201

HTH

Adam

"Detecting" whether a neutral cycle exists or not will be difficult in general.

However, when you do know that there is a parabolic cycle (e.g. because you have constructed your map that way), Braverman shows, as quoted by Adam, that you can compute the Julia set in polynomial time. What should be noted here is that the program used (and the time bound) will depend on the parameter in question.

For irrationally indifferent fixed points, things can get even more tricky. But when the rotation number is nice (e.g. some quadratic irrational such as the 'golden mean'), you will have a critical point on the boundary of the Siegel disk. So you can draw the boundary of the Siegel disk by plotting the orbit of the critical point.

In parameter spaces, rather than trying to 'detect' neutral cycles, you may wish to draw the boundaries of hyperbolic components using Newton's method. That is, take a point in the hyperbolic component that you are interested in (where there is an attracting cycle), and then find a curve along which the modulus of the multiplier tends to one. Then you will have found an indifferent parameter. Now you can similarly change the argument of the multiplier, again using Newton's method, and trace this curve. Some care is required near "cusps".

For an example of such a picture, in the family of exponential maps, see Figure 1 in my article with Dierk Schleicher, "Bifurcations in the space of exponential maps" (http://arxiv.org/abs/math/0311480).