
## 1. Density results for PCF polynomials

Assume $f(z)\in \mathbb{Q}[z]$ is PCF (orbit of critical points is finite) and some reasonable starting point $\alpha\in\mathbb{Q}.$
1. #### Problem 1.1.

[Xander Faber] Describe $N(x)=\#\{p\leq x : p|f^n(\alpha) \text { for some } n\}$ for $p$ prime as $x\rightarrow \infty.$
1. Remark. There are results for non PCF maps.
• Remark. The guess here is that $N(x)=o(\pi(x)).$
• #### Problem 1.2.

[Joe Silverman] What is the growth rate of $N(x)$ for PCF and non PCF maps?
• #### Problem 1.3.

If $f(z)\in\mathbb{F}_q(T)$, what happens then? What happens with other fields?

Cite this as: AimPL: The Galois theory of orbits in arithmetic dynamics, available at http://aimpl.org/galarithdyn.