  Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say
that an endomorphism $\Phi\colon \mathbb{P}_1\to \mathbb{P}_1$ has good
reduction 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 reduction
that 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, the
simple and the critically good reduction. The last part of our article is
dedicated to prove a characterization of the maps whose iterates, in a certain
sense, preserve the critically good reduction.

Number Theory;Algebraic Geometry;Dynamical Systems