Crucial curvatures and minimal resultant loci for non-archimedean polynomials

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 $P$.Comment: 29 pages, 1 figur

Böttcher coordinates at wild superattracting fixed points

Let (Formula presented.) be a prime number, let (Formula presented.) with (Formula presented.), and let (Formula presented.) be the Böttcher coordinate satisfying (Formula presented.). Salerno and Silverman conjectured that the radius of convergence of (Formula presented.) in (Formula presented.) is (Formula presented.). In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result