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 . [ABSTRACT FROM AUTHOR]