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

Abstract

We introduce the $f$-crucial function ${\mathrm{Crucial}}_f$ 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 ${\mathrm{ordRes}}_f$ 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 ${\mathsf{P}}^1$.

39 pages. (v4) Final version. (v3) The title is modified to be more informative. An effective estimate of the size of the $f$-crucial set is added. Some typos are corrected. (v2) An application to dynamics is added

References in corpus (4)

• The Geometry of the Minimal Resultant Locus
• Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics
• An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line
• Iteration and the Minimal Resultant