# infinitesimal classification of functions near a fixed point upto conjugation

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?

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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.