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# Crucial curvatures and minimal resultant loci for non-archimedean polynomials

Cornell University - arXiv
May 2020

Let $K$ be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value. For a polynomial $P\in K[z]$ of degree $d>1$, the $n$-th Trucco's dynamical tree $\Gamma_n$, $n\ge 0$, is spanned by the union of $P^{-n}(\xi_B)$ and $\{\infty\}$ in the Berkovich projective line over $K$, where $\xi_B$ is the boundary point of the minimal Berkovich closed disk in the Berkovich affine line containing the Berkovich filled-in Julia set of $P$. We expand Trucco's study on the branching of $\Gamma_n$ and, using the second author's Berkovich hyperbolic geometric development of Rumely's works in non-archimedean dynamics and on their reductions, compute the weight function on $\Gamma_n$ associated to the $\Gamma_n$-crucial curvature $\nu_{P^j,\Gamma_n}$ on $\Gamma_n$ induced by $P^j$, for $j\ge 1$ and $n\ge 1$. Then applying Faber's and Kiwi and the first author's depth formulas to determine the GIT semistability of the coefficient reductions of the conjugacies of $P^j$, we establish the Hausdorff convergence of the barycenters $(\operatorname{BC}_{\Gamma_n}(\nu_{P^j,\Gamma_n}))_n$ towards Rumely's minimal resultant locus $\operatorname{MinResLoc}_{P^j}$ of $P^j$ in the Berkovich hyperbolic space and the independence of $\operatorname{MinResLoc}_{P^j}$ on $j\ge d-1$. We also establish the equidistribution of the averaged total variations $(|\nu_{P^j,\Gamma_n}|/|\nu_{P^j,\Gamma_n}|(\Gamma_n))_n$ towards the $P$-equilibrium (or canonical) measure $\mu_P$, for an either nonsimple and tame or simple $P$.

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