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Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We saythat an endomorphism $\Phi\colon \mathbb{P}_1\to \mathbb{P}_1$ has goodreduction at $v$ if there exists a model $\Psi$ for $\Phi$ such that$\deg\Psi_v$, the degree of the reduction of $\Psi$ modulo $v$, equals$\deg\Psi$ and $\Psi_v$ is separable. We prove a criterion for good reductionthat is the natural generalization of a result due to Zannier in \cite{Uz3}.Our result is in connection with other two notions of good reduction, thesimple and the critically good reduction. The last part of our article isdedicated to prove a characterization of the maps whose iterates, in a certainsense, preserve the critically good reduction.

Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus