$S$-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 $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $\alpha \in \mathbb{P}^1(K)$ non-preperiodic and $\beta \in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+\epsilon})$ on the number of points in the forward orbit of $\alpha$ that are $S$-integral relative to $\beta$, extending results of Hsia--Silverman [HS11]. When $\varphi$ is a power or Latt`es map, we obtain a stronger upper bound of the form $O(|S|)$. We also prove uniform bounds when $\varphi$ is a polynomial, extending results of Krieger et al [KLS+15].