On the number of critical points in parabolic basins

Abstract

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. 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\mathbb{C} also exists. 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 \textrm{Re}\alpha\leq \frac14m+\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.