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Let $K$ be a number field and let $\phi$ in $K(z)$ be a rational function ofdegree $d\geq 2$. Let $S$ be the places of bad reduction for $\phi$ (includingthe archimedan places). Let $Per(\phi,K)$, $PrePer(\phi, K)$, and$Tail(\phi,K)$ be the set of $K$-rational periodic, preperiodic, and purelypreperiodic points of $\phi$, respectively. The present paper presents two mainresults. The first result gives a bound for $|PrePer(\phi,K)|$ in terms of thenumber of places of bad reduction $|S|$ and the degree $d$ of the rationalfunction $\phi$. This bound significantly improves a previous bound given by J.Canci and L. Paladino 2014. For the second result, assuming that $|Per(\phi,K)|\geq 4$ (resp. $|Tail(\phi,K)| \geq 3$), we prove bounds for $|Tail(\phi,K)|$(resp. $|Per(\phi,K)|$) that depend only on the number of places of badreduction $|S|$ (and not on the degree $d$). We show that the hypotheses ofthis result are sharp, giving counterexamples to any possible result of thisform when $|Per(\phi,K)| < 4$ (resp. $|Tail(\phi,K)| < 3$).

Bound for preperiodic points for maps with good reduction