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

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 $\mathsf{P}^1$.