The Geometry of the Minimal Resultant Locus

Abstract

Let K be a complete, algebraically closed, nonarchimedean valued field, and let f(z) be a rational function in K(z) of degree d at least 2. We show there is a natural way to assign non-negative integer weights w_f(P) to points of the Berkovich projective line over K, in such a way that the sum over all points is d-1. When f(z) has bad reduction, the set of points with nonzero weight forms a distributed analogue of the single point which occurs when f(z) has potential good reduction. Using this, we characterize the Minimal Resultant Locus of f(z) in dynamical and moduli-theoretic terms: dynamically, it is the barycenter of the weight-measure attached to f(z); moduli-theoretically, it is the closure of the set of type II points where f(z) has semi-stable reduction in the sense of Geometric Invariant Theory.

53 pages

References in corpus (4)

• Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps
• The Geometry of the Minimal Resultant Locus
• Sur la structure des ensembles de Fatou p-adiques
• An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line

Cited by in corpus (10)

• The Geometry of the Minimal Resultant Locus
• An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line
• Iteration at the Boundary of Newton Maps
• Iteration and the Minimal Resultant
• The Lipschitz Constant of a Nonarchimedean Rational Function
• Indeterminacy loci of iterate maps in moduli space
• Compactifications of the moduli spaces of Newton maps
• Counting rational maps on ${\mathbb{P}}^1$ with prescribed local conditions
• Geometric formulas on Rumely's weight function and crucial measure in non-archimedean dynamics
• Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics