how many known classifications and types of indifferent fixed-points are there?

Question

In complex dynamics, the behavior of a fixed point $z_0$ is characterized by the value of the derivative at that point. We say that $z_0$ is

• Attractive if $|f'(z_0)|<1$,
• Repulsive if $|f'(z_0)|>1$, or
• Indifferent if $|f'(z_0)|=1$

The third case is much more subtle than the other two and it appears there are several possible behaviors for the dynamics near such a point. If $f'(z_0)$ is a root of unity then the Leau-Fatou flower theorem describes the dynamics.

How many known classifications and types of indifferent fixed-points are there?

Answer 1

There are essentially three different situations as described in sections 6.5, 6.6, and 6.7 of Alan Beardon's Iteration of Rational functions.

• A parabolic point, where the multiplier is a root of unity. An example is $f(z)=z+z^5$ which has a parabolic fixed point at the origin and whose Julia set is shown in the first figure below.
• An irrationally indifferent point in the Fatou set. For an indifferent fixed point, this happens if and only if the function is analytically conjugate to a linear function and the component of the Fatou set containing the point is called a Siegel disk. An example is $$f(z)=e^{2\pi i \varphi}z+z^2,$$ where $\varphi$ is the golden ratio. The Julia set is shown in the second figure below.
• An irrationally indifferent point in the Julia set. This type is not well understood. At least some of these are known to be uncomputable in polynomial time and, as far as I know, no accurate pictures have been generated of any Julia set in this class.

A parabolic fixed point for $f(z)=z+z^5$

A Siegel disk for $f(z)=e^{2\pi i \varphi}z+z^2$