Vol. 11, No. 4, 2017

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A new equivariant in nonarchimedean dynamics

Robert Rumely

Vol. 11 (2017), No. 4, 841–884

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.

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