Morphisms of Berkovich curves and the different function

Given a generically étale morphism f:Y→X of quasi-smooth Berkovich curves, we define a different function δ_f:Y→[0,1] that measures the wildness of the topological ramification locus of f. This provides a new invariant for studying f, which cannot be obtained by the usual reduction techniques. We prove that δ_f is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δ_f can be used to explicitly construct the simultaneous skeletons of X and Y. As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p.