The average number of integral points in orbits

Over a number field $K$, a celebrated result of Silverman states that if $\phi (z)\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit ${\text{Orb}}_{\phi }(P)=\{{\phi }^n(P){\}}_{n\ge 0}$ is finite for all $P\in $\mathbb{P}$^1(K)$\mathrm{.}Inthispaper,weshowthatifwevary$\varphi$and$P$ in a suitable family, the number of $S$-integral points in ${\text{Orb}}_{\phi }(P)$ is absolutely bounded. In particular, if we fix $\phi$ and vary the basepoint $P\in {\mathbb{P}}^1(K)$, then we show that $\mathrm{\#}({\text{Orb}}_{\phi }(P)\cap {\mathcal{O}}_{K,S})$ is zero on average. Finally, we prove a zero-average result in general, assuming a standard height uniformity conjecture in arithmetic geometry.