## Scarcity of finite orbits for rational functions over a number field

### Volume 190 / 2019

#### Abstract

Let $\phi $ be an endomorphism of degree $d\geq {2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\phi $ has good reduction outside $S$. The article presents two main results. The first result is a bound on the number of $K$-rational preperiodic points of $\phi $ in terms of the cardinality of $S$ and the degree $d$ of $\phi $. This bound is quadratic in $d$, which is a significant improvement to all previous bounds in terms of $d$. The second result is that if there is a $K$-rational periodic point of period at least 2, then there exists a bound on the number of $K$-rational preperiodic points of $\phi $ that is linear in $d$.