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arXiv · 1612.04441

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

Yûsuke Okuyama
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Abstract

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 $ρ$ 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 $ν_f$ 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 $(ν_{f^n})_n$ of $f^n$-crucial measures towards the $f$-equilibrium (or canonical) measure $μ_f$ on $\mathsf{P}^1$.

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Yûsuke Okuyama. 2019-07-30. Geometric formulas on Rumely's weight function and crucial measure in non-archimedean dynamics. https://arxiv.org/abs/1612.04441

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