infinitesimal classification of functions near a fixed point upto conjugation

Question

Let f(x) be a power series with complex coefficients, suppose f(0)=0. Is there a classification of equivalence classes of f(x) up to conjugation by another power series g(x) with g(0)=0? This question should be interesting in the field of complex analytic dynamical systems, since conjugate functions define similar dynamics. By considering the equations of the coefficients of a conjugate series, it can be shown that unless f'(0) is 0 or some n-th root of unit, f(x) is conjugate to a linear function. I am wondering if there are known results to other cases?

Answer 1

This question is surely very interesting in holomorphic dynamics. A lot is already known, some of it going back to Koenigs (around 1880). As you mentioned, a germ $f(x)= a_{1}x+a_{2}x^2 + \ldots $ with $\vert a_{1} \vert \neq 0,1$ is holomorphically conjugated to its linear part $g(x)=a_{1}z$.

When the germ is tangent to the identity (i.e $a_1 =1$), the classification is more complicated, but is still possible (results due to Ecalle and Voronin). The case where $a_{1}=e^{2 \pi i p /q}$ is similar to the case $a_1=1$.

When $a_{1}=e^{2 \pi i \theta}$ with $\theta \notin \mathbb{Q}$, the germ $f$ is only formally conjugated to its linear part. We say that the origin is a Siegel point if the germ is locally linearizable, and is a Cremer point otherwise. Here Bryuno, Yoccoz, Perez-Marco made essential contributions to the study of such points. Milnor's book on Complex dynamics has a very clear chapter on that.

Also on the arxiv, M. Abate has several very helpful surveys about Discrete local holomorphic dynamics, including one about open questions in the domain.

The theory in the higher dimensional setting is even more complicated.