Fixed point proportions for Galois groups of non-geometric iterated extensions

Abstract: Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow \mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})$ for each prime $\mathfrak{p}$ of good reduction, induced by reduction modulo $\mathfrak{p}$. It has been shown that for a typical $\varphi$ the proportion of periodic points of $\varphi_\mathfrak{p}$ should tend to $0$ as $|\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})|$ grows. In this paper, we extend previous results to include a weaker set of sufficient conditions under which this property holds. We are also able to show that these conditions are necessary for certain families of functions, for example, functions of the form $\varphi(x)=x^d+c$, where $0$ is not a preperiodic point of this map. We study the proportion of periodic points by looking at the fixed point proportion of the Galois groups of certain extensions associated to iterates of the map.