Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 4

The holomorphic fixed-point index, or residue index, isFor a simple fixed point with multiplier $\lambda \ne 1$ it equals $1/(1-\lambda )$. Three distinct fixed points of a quadratic rational map are simple, and the rational fixed-point formula givesIf, say, ${\lambda }_0{\lambda }_1=1$, thenThe remaining term would have to be zero, which is impossible. Thus the multiplier relation for three distinct fixed points of a quadratic rational map is

The parabolic basin $A$ is open and lies in $F(g)$. Suppose $z\in \partial A\cap F(g)$ and let $U$ be the Fatou component containing $z$. A small neighborhood of $z$ in $U$ meets $A$. On one point of $U$, the iterates eventually enter the chosen attracting petal and converge to zero along its attracting vector. Normality and the identity theorem for the limiting iterates make the same true throughout the connected component $U$. Hence $U\subseteq A$, which is impossible for a boundary point. Therefore the parabolic basin satisfies

Let $p$ be prime and defineThe recursion $P_{n+1}=P_n^2+c$ shows that $P_p$ has degree $2^{p-1}$. Moreover $P_p(c)=c+O(c^2)$ at zero, so $c=0$ is a simple root. Since the degree is greater than one, $P_p$ has a nonzero root $c_0$. At $c_0$, the critical point zero is periodic with period dividing $p$. It is not fixed because $c_0\ne 0$, so primality makes its exact period $p$. Thus $c_0$ is the center of a hyperbolic component of exact period $p$.

The multiplier map on this component covers the unit disc. Move to its boundary along parameters whose attracting-cycle multiplier tends to $-1$. Compactness of the Mandelbrot set gives a limiting parameter $c_{\ast }$. The periodic cycle persists with exact period $p$: at multiplier $-1$, every point is a simple root of $f_{c_{\ast }}^p(z)-z$, so no collision to a lower-period orbit occurs. Its multiplier is the root of unity $-1$, and hence it is a parabolic cycle after squaring the return map. We have produced a parabolic cycle of exact period $p$ for every prime $p$. Therefore the parabolic periods in the quadratic family form an infinite set.