Preperiodic points for rational functions defined over a global field in   terms of good reductions
  Let $\phi$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an $S$-unit equation theorem in 2 variables. 
Jung-Kyu Canci and Laura Paladino
math.NT math.DS