00-16   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## On the number of critical points in parabolic basins

Let $f$ be a rational function with a fixed point $z_0$ of multiplicity $m+1$. Then there are $m$ invariant components of the Fatou set, called immediate parabolic basins, where the iterates of $f$ tend to $z_0$, and each immediate parabolic basin contains at least one critical point of $f$. Also, there exists a function $\phi$ holomorphic and univalent near $z_0$ with $\phi(z_0)=0$ such that $h:=\phi\circ f\circ\phi^{-1}$ satisfies $h(z)=z-z^{m+1}+\alpha z^{2m+1}+O(z^{2m+2})$ as $z\to 0$ for some $\alpha\in\C$. This number $\alpha$ is invariant under holomorphic changes of coordinates. We show that if each immediate parabolic basin at $z_0$ contains exactly one critical point of $f$, and if this critical point is simple, then $\re\alpha\leq \frac{m}{4}+\frac12$. We also discuss the case that the critical points in the immediate parabolic basins are multiple and establish an upper bound for the real part of $\alpha$ depending on $m$ and the multiplicities of the critical points contained in the parabolic basins.

Mathematics Subject Classification (1991): Primary 37F10; Secondary 30D05, 37F45

Bibliographical note: Ergodic Theory Dynam. Systems 22, 655-669 (2002).