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I've been digging around the iteration of rational functions. By chance I came across a mapping $R(z)$ such that there is only one reppelling fixed point and two parabolic points. I'm still far from understading the texts in the field, e.g. Beardon's or the classical text by Milnor, but all mention that the study of a mapping with parabolic points is significantly more difficult than the case of attracting and reppelling fixed points.

Can someone give, perhaps a heuristically, explanation why this analysis is more difficult? In particular, I would like to understand if the dynamics depends explicitly on the mapping (in which case there's nothing much to do, I guess), or if it's really "difficult" in the sense of the complexity of proving theorems about the dynamics.

Examples will be much appreciated.

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First off, let's be clear on the definition of a parabolic point. In the study of the iteration of an analytic function $f:\mathbb C \to \mathbb C$ we say that a fied point $z_0$ is parabolic if $f'(z_0)$ is a root of unity. In particular, $|f'(z_0)|=1$ so that $f$ is neither repulsive ($|f'(z_0)|>1$) or attractive ($0<|f'(z_0)|<1$) at $z_0$ or super-attractive ($f'(z_0)=0$).

If $z_0$ is either a repulsive or attractive fixed point of $f$, then it turns out that $f$ is dynamically similar near $z_0$ to $g(z)=az$ near zero. In particular, there is a neighborhood of $z_0$ where all points move either away or toward $z_0$ under iteration of $f$. A similar statement can be made when $z_0$ is super-attractive, though that is treated a bit differently.

When we iterate near a parabolic point, we simply don't have that classification. There are at least two concrete properties that we lose that makes analysis more difficult:

  1. There's no longer a neighborhood where we move either away or toward $z_0$ under iteration of $f$. In fact, there are always some directions where we move towards $z_0$ and other directions where we move away. The exact description of those directions is the content of the Leau-Fatou flower theorem.
  2. Points move very slowly under iteration near a parabolic fixed point, which makes computer generation of pictures very time consuming.

A relatively simple example illustrating all this is given by the function $f(z) = z+z^5$. Note that zero is a parabolic fixed point for this function. If $|r|$ is small, then $|r|^5$ will be very small. Thus, points near zero barely move under application of $f$. This illustrates point 2 above.

To illustrate point 1 above, choose a complex number $z$ in the polar form $z=re^{n\pi i/4}$. Then $$ f(z) = re^{n\pi i/4} + r^5 e^{5n\pi i/4}. $$ Note that when $n$ is even, the displacement $z^5$ is in the direction of $z$ and, thus, away from the origin. When $n$ is odd, the displacement $z^5$ is in the opposite direction of $z$ and, thus, toward the origin.

The dynamics of $f$ are illustrated in the following picture:

enter image description here

Returning to point 2, the basic escape time algorithm is not particularly good at generating images of these types of Julia sets. A solid understanding of the Leau-Fatou flower theorem, however, allows us to to classify the dynamics based on the repelling/attracting directions. We can then use a boundary scanning technique to generate the following:

enter image description here

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  • $\begingroup$ Do you have a reference on how to construct the boundary scanning technique? $\endgroup$ Oct 18 '19 at 21:10
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    $\begingroup$ I discussed this a little bit in my answer to this question. As I mentioned there, it's really quite a common technique in image manipulation. It's built into photoshop, for example. $\endgroup$ Oct 18 '19 at 21:13
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    $\begingroup$ commons.wikimedia.org/wiki/File:Julia_set_z%2Bz%5E5.png $\endgroup$
    – Adam
    Oct 20 '19 at 5:35
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( The answer by Mark is great)

Le'ts take simple example: discrete complex dynamical system with

  • one parabolic fixed point with multiplicity p > 1 ( = p sepals)
  • one parameter
  • fixed point moved to zero

The global dynamics is similar to other cases.

The local dynamics ( near fixed point) is more complicated then other cases because of:

  • slow dynamics - escape time is 2^n in parabolic case ( exponential slowdown) and n in the hyperbolic case
  • complicated move of point near fixed point

Let's divide move into:

  1. angular move = rotation around fixed point( see doubling map)
  2. radial move ( see external and internal rays, invariant curves ) fallin into target set and attractor ( in hyperbolic and parabolic case )

To do it one have divide orbit of point into p suborbits ( each orbit is inside one sepal parabolic Julia set with 2 sepals. One orbit ( white dots ) was plotted to check invariant curves

Now after 3 steps one have p simple orbits ( suborbits). Take one orbit and you can analyze it. It can be seen on the above image as the white dots. Note that all dots are on the same color of the sepal, it means that point with the same color are on the invariant curve.

Here is analysis of radial move only:

Parabolic case is red: first point go out of fixed point then goes back to the fixed point

Parabolic case is red: first point go out of fixed point then goes back to the fixed point

This is analysis of (sub)orbit of point near fixed point.

There is also different approach: analyze behaviour of periodic ponts.

It is based not on the static image ( one Julia set) but changes in Julia set when parameter is moved on the parameter plane

enter image description here

Here oner can see how period p=2 repelling cycle ( landing points of external ray) is approaching fixed point.

At the end it collapse into the fixed point.

Now ( period 2 = fixed ) is the parabolic dynamics.

More one can find in wikibooks: * Parabolic dynamics

HTH

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