Metric uniformization of morphisms of Berkovich curves

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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=(ΓY,Γₓ) of f, the sets Nf,≥ ₙ of points of Y of multiplicity at least n in the fiber are radial around ΓY with the radius changing piecewise monomially along ΓY. In this case, for any interval l=[z,y]⊂ Y connecting a rigid point z to the skeleton, the restriction f|ₗ gives rise to a profile piecewise monomial function φy [0,1]→[0,1] that depends only on the type 2 point y∈ΓY. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φy} with y∈ΓY⁽²⁾. We prove that this family is piecewise monomial in y and naturally extends to the whole Yhyp. In addition, we extend the theory of higher ramification groups to arbitrary real-valued fields and show that φy coincides with the Herbrand's function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand's function, which applies also to non-normal and even inseparable extensions.