Vol. 11, No. 4, 2017

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A new equivariant in nonarchimedean dynamics

Robert Rumely

Vol. 11 (2017), No. 4, 841–884
Abstract

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) K(z) have degree d 2. We show there is a canonical way to assign nonnegative integer weights wφ(P) to points of the Berkovich projective line over K in such a way that Pwφ(P) = d 1. When φ has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when φ has potential good reduction. Using this, we characterize the minimal resultant locus of φ in analytic and moduli-theoretic terms: analytically, it is the barycenter of the weight-measure associated to φ; moduli-theoretically, it is the closure of the set of points where φ has semistable reduction, in the sense of geometric invariant theory.

Keywords
dynamics, minimal resultant locus, crucial set, repelling fixed points, nonarchimedean weight formula, geometric invariant theory
Mathematical Subject Classification 2010
Primary: 37P50
Secondary: 11S82, 37P05
Milestones
Received: 27 August 2015
Revised: 13 January 2017
Accepted: 11 February 2017
Published: 18 June 2017
Authors
Robert Rumely
Department of Mathematics
University of Georgia
Athens, GA 30602
United States