We introduce the $f$-crucial function $\operatorname{ Crucial}_f$ associated to a rational function $f\in K(z)$ 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 $\operatorname{ ordRes}_f$ in terms of the hyperbolic metric $\rho$ on the Berkovich upper half space $\mathsf{ H}^1$ in the Berkovich projective line $\mathsf{ P}=\mathsf{ P}(K)$. We also obtain geometric formulas for Rumely's weight function $w_f$ and crucial measure $\nu_f$ on $\mathsf{ P}^1$ 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 $(\nu_{ f^n})_n$ of $f^n$-crucial measures towards the $f$-equilibrium (or canonical) measure $\mu_f$ on $\mathsf{ P}^1$.
Categories: math.NT math.DS