Let $f(z)$ be an entire function of order $0$ with at least $3$ zero’s.
$$ f(z) = a_0(1 - a_1 z)(1 - a_2 z)(1 - a_3 z)... $$
There are 2 cases :
A) $f(z)$ is polynomial of degree $3$ or higher. B) $f(z)$ has infinitely many zero’s.
When looking at julia sets I noticed polynomials of degree $3$ or higher tend to have an attracting fixpoint ? Or is that an illusion of me and did i only look at those because they have nicer Julia sets ?
So this Made me wonder about case A. Also let us say I want to compute how many attracting fixpoints a degree 7 polynomial has. For me , not knowing any tricks , that is hard ?! So first i would compute the zero’s of a 7th degree polynomial ( the original one minus the identity $z$ ; to find the fixpoints ) , despite not having a closed form method nor closed form solution ( usually ). Then I plug in those values $a_i$ in the derivative of the ( original ) 7th degree polynomial : $ f ‘ (a_i) = b_i $. And then i compute $Re(\ln(b_i))$ to see if the fixpoint is attracting or not. And then I list the attracting ones and count how many there are. I could perhaps come up with a kind of calculus way like integrals and contour integrals, but i doubt that would be much easier ( for me ? ). Although perhaps a very efficient way to use calculus exist. And maybe Some theorems might exist that help. I thought about the sum and products of the zero’s and their effect and similar ideas. But I was not able to get far. Besides i prefer to stick with algebra.
I know the zero’s of $f’(z)$ relate to those of $f(z)$ by a theorem of Gauss-Lucas ( that polygon one ). But I also failed to theoretically or practically use that theorem as well.
Dynamical systems texts usually focus on a specific fixpoint i assume, anyways reading them did not help me either.
Also I am not sure what the relevance of numerical stability would be.
Short story : I barely know what I am doing or at least feel that way.
As for the case $B$ we have the same problems ofcourse but since it is not a polynomial different methods , theorems and conclusions might apply.
I know the smallest fixpoints of $exp(z)$ are repelling but note that is not of type $B$. Also I believe $exp$ has attracting fixpoints as well, right ?
Actually I rarely read about functions of type $B$ and when I do they are usually complicated functions with open problems ( RH type usually ). Anyways , just pointing out I lack much experience with functions of type $B$.
I assume the case $B$ is harder then the case $A$.
So how to compute the number of attracting fixpoints efficiently and what can be said about the results ??
Or even how to estimate them ? Lower bound ? Upper bound ? Probability ? Patterns ?
With appropriate software, it's easy to numerically find the fixed points and evaluate the absolute value of the derivative at those fixed points. Any of the usual CAS's (Maple, Mathematica, etc) should be able to do it.
I don't think there's any sense in the statement that polynomials of degree $3$ or higher "tend to have an attracting fixpoint", unless you specify a probability distribution for your polynomials.
As for $\exp$, its fixed points are the branches of $-W(-1)$, where $W$ is the Lambert W function. Since the derivative of $\exp$ is $\exp$, and all these fixed points have absolute value $> 1$, they are all repelling.
EDIT: Suppose we look at polynomials of the form $f(z) = z + c \prod_{j=1}^d (z - r_j)$. This has degree $d$ and fixed points $r_1, \ldots, r_j$ (which I'll assume are distinct). Note that $f'(r_j) = 1 + c \prod_{i\ne j} (r_j - r_i)$. To make a polynomial with fixed points $r_j$, none of which are attracting, just take $c$ large enough. If the $r_j$ are distinct integers and $n \ge 5$, $c = 1$ will do.