Geometric formulas on Rumely’s weight function and crucial measure in non-archimedean dynamics

We introduce the f -crucial function Crucial f associated to a rational function f ∈ K ( z ) of degree > 1 over an algebraically closed field K of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely’s (resultant) function ordRes f in terms of the hyperbolic metric ρ on the Berkovich upper half space H 1 in the Berkovich projective line P 1 = P 1 ( K ) . We also obtain geometric formulas for Rumely’s weight function w f and crucial measure ν f on P 1 associated to f , as well as improvements of Rumely’s principal results. As an application to dynamics, we obtain a quantitative equidistribution of the sequence ( ν f n ) n of f n -crucial measures towards the f -equilibrium (or canonical) measure μ f on P 1 .