Integral points of bounded degree on the projective line and in dynamical orbits

Abstract

Let $D$ be a non-empty effective divisor on ${\mathbb{P}}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on ${\mathbb{P}}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\phi (z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate ${\phi }^2(z)$ is not a polynomial. We show that as we vary over points $P\in {\mathbb{P}}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.