42 research outputs found
Crucial curvatures and minimal resultant loci for non-archimedean polynomials
Let be an algebraically closed field that is complete with respect to a
non-trivial and non-archimedean absolute value. For a polynomial of
degree , the -th Trucco's dynamical tree , , is
spanned by the union of and in the Berkovich
projective line over , where is the boundary point of the minimal
Berkovich closed disk in the Berkovich affine line containing the Berkovich
filled-in Julia set of . We expand Trucco's study on the branching of
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 associated to the
-crucial curvature on induced by
, for and . 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 , we establish the Hausdorff convergence
of the barycenters
towards Rumely's minimal resultant locus of
in the Berkovich hyperbolic space and the independence of
on . We also establish the
equidistribution of the averaged total variations
towards the
-equilibrium (or canonical) measure , for an either nonsimple and
tame or simple .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