Illustrating the Leau-Fatou flowers

[MphLee]

Excuse me, I'm not lazy or what... only very busy atm... but I'd like to begin to chew this material... everyone here seems to understand terms and definitions but I'm a total beginner.
What the hell is, formally, a flower? Looking at the pictures it seems the image of a path. In particular it seems it has to do with images of the family of paths \[\gamma_n:\mathbb S^1\to \mathbb C\]
defined as \(\gamma_n:=f^n\circ \gamma_0\) where \(\gamma_0\) is the is the inclusion of the circle as the unit circle, or a scaled version of it.

Question 1: what this family of loops/paths has to do with the formal definition of flowers and petals?

Question 2: in any case, from the algebraic topology pov, each of these \(\gamma_n\) can be sent to their homotopy-class \([\gamma_n]\in \pi_1(\mathbb C)\). In the case of the complex plane, has the behavior of the induced discrete action on the fundamental group of the complex plane being studied? In general, if \(X\) is a topological space equipped with a continuous map \(f:X\to X\), and \(p\in {\rm fix}(f)\), we can define by post-composition an endomap \[{\bar f}:\pi_1(X;p)\to\pi_1(X;p)\] (for a proof use functoriality the first-homotopy group construction).
The question is, what the induced dynamics on the homotopy groups tell us about the original dynamics?

[bo198214]

but I'd like to begin to chew this material... everyone here seems to understand terms and definitions but I'm a total beginner.
What the hell is, formally, a flower?

You can find a proper definition of a petal in Milnor in the beginning of §7.
A flower is just all the petals of a fixed point.

Looking at the pictures it seems the image of a path. In particular it seems it has to do with images of the family of paths \[\gamma_n:\mathbb S^1\to \mathbb C\]

No, nothing like that - it is just an *illustration*! But with this illustration one can well see the attracting and repelling petals.

[MphLee]

Thank you. Now I get what you mean by illustration. Those paths' deformation is caused by the flower. It is perfectly clear now.

[tommy1729]

Just to give some visual impressions I was drawing some Leau-Fatou flowers.
Leau-Fatou flowers occur at parabolic fixed points (i.e. \(f'(z_0)^n=1\) for some integer n>0).
Here we look at the most prominent case \(f'(0)=1\) for the most simple functions, i.e. polynomials.
The Leau-Fatou flower has 2*p petals, where p is the iterative value of f (Ecalle writes it \(p={\rm valit}[f]\)),
which means the index k in the powerseries development of f up to which it is the identity function.
For example \({\rm valit}[z\mapsto z+z^2]=1\), \({\rm valit}[z\mapsto z+z^{10}]=9\).
The petals alternating in attracting/repelling when moving around the fixed point.

So I took a circle around the fixed point and see how it deformed under application of f.
These are the pictures


On each petal there is an Abel function or called Fatou-Coordinate defined, and these functions are typically different on each petal (except the logit is analytic at the fixed point, but in case of polynomials (and entire functions) this is not the case).
The same applies to fractional iterates.
That's why it is interesting to look at the Borel-Summation of the formal powerseries logit, or the formal powerseries of some iterative root.
The Borel-Summation should provide values on a circle around the fixed point, while the circle traverses the different petals.
So if the Borel-summation of a (say) a half iterate returns the a different function on each petal, then there should be breaks visible between the petals.
Maybe the Borel-summation returns only one function and its continuation around the circle - then one would need to find different ways to obtain the other half-iterates.

Just for comparison I put here the Leau-Fatou flower of a function with analytic logit, which has a half-iterate that is analytic at 0.
This function is \(f(x)=-{\rm arccot}(1-{\rm cot}(x))\) it has this power series development at 0:
\[f(x)=x +  x^{2} +  x^{3} + \frac{2}{3} x^{4} - \frac{43}{45} x^{6} - \frac{29}{15} x^{7} - \frac{778}{315} x^{8} - \frac{374}{189} x^{9} + \frac{122}{14175} x^{10} + \frac{782}{225} x^{11} + \frac{3515884}{467775} x^{12} + \frac{45094}{4455} x^{13} + \frac{360268549}{42567525} x^{14} - \frac{1110241}{19348875} x^{15} + O(x^{16}) \]
So it has two petals, but it looks more "round" or convex than the flower of \(z+z^2\):

Also just for comparison the flower of the sine \(x - \frac{1}{6} x^{3} + \frac{1}{120} x^{5} - \frac{1}{5040} x^{7} + \frac{1}{362880} x^{9} + O(x^{10})\) and \({\rm valit}(\sin)=2\) so it has 4 petals:


As final picture "the growth of the flower"   I continuously increase p in \(z+z^p\):

Unfortunately there is a typo in the title, I forgot the r in \(\gamma\) but you know, what I meant!

So hopefully soon, I will continue with the calculation of the Borel-summation.

About those 2 first pictures.

The curve iterations intersect the initial curve at some points.

Those points seems to converge to some values.

So what are those values for say x + a * x^p + b * x^q ?

We know the number of intersections for x + x^p are 2(p-1) but where do the limits go to ?

Are those numbers fixpoints of something or so ?

Are they easy to compute ?

Do they have a closed form ?


Regards

tommy1729

[MphLee]

Should those points converge on the boundary of the respective petals?
I don't think we can say they are part of the same orbit.
Assume the initial path is the unitary circle. Those intersection points are of the general form
\[\{ p\in \mathbb S^1\, |\, \exists n\in \mathbb N. \exists \theta\in [0,2\pi).\, f^n(e^{i\theta})=p\}\]

Maybe to each \(\theta\) we could associate/study the set of \(n\) such that \(|f^n(e^{i\theta})|=1\).
Not sure how this can be useful.