A new equivariant in nonarchimedean dynamics

Rumely, Robert

doi:10.2140/ant.2017.11.841

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A new equivariant in nonarchimedean dynamics

Robert Rumely

Vol. 11 (2017), No. 4, 841–884

DOI: 10.2140/ant.2017.11.841

Abstract

Let $K$ be a complete, algebraically closed nonarchimedean valued field, and let $φ (z) \in K (z)$ have degree $d \geq 2$ . We show there is a canonical way to assign nonnegative integer weights $w_{φ} (P)$ to points of the Berkovich projective line over $K$ in such a way that $\sum_{P} w_{φ} (P) = d - 1$ . 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

Vol. 11, No. 4, 2017