$S$-Integral Points in Orbits on ${\mathbb{P}}^1$

Abstract

Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $\ddot{I}†:{\mathbb{P}}^1\to {\mathbb{P}}^1$ be a rational map of degree $d\ge 2$ defined over $K$. Given $\hat{I}\pm \in {\mathbb{P}}^1(K)$ non-preperiodic and ${\hat{I}}^2\in {\mathbb{P}}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S{|}^{1+\hat{I}\mu })$ on the number of points in the forward orbit of $\hat{I}\pm$ that are $S$-integral relative to ${\hat{I}}^2$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $\ddot{I}†$ is a polynomial, extending resaults of Krieger et al [KLS+15].

Siginificantly shortened following reviewer's comments