On some notions of good reduction for endomorphisms of the projective line

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Let Φ be an endomorphism of (), the projective line over the algebraic closure of, of degree ≥2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds for any pair of branch points. We say that Φ has simple good reduction at v if the map Φᵥ, the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φᵥ is separable, then Φ has simple good reduction at v.