Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms f:Y→X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ=(Γ_Y,Γ_X) of f, the sets N_f,≥n of points of Y of multiplicity at least n in the fiber are radial around Γ_Y with the radius changing piecewise monomially along Γ_Y. In this case, for any interval l=[z,y]⊂Y connecting a point z of type 1 to the skeleton, the restriction f|_l gives rise to a profile piecewise monomial function φ_y:[0,1]→[0,1] that depends only on the type 2 point y∈Γ_Y. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φ_y} with y∈Γ_Y ⁽²⁾. We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φ_y coincides with the Herbrand function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.