Semistable intrinsic reduction loci for the iterations of non-archimedean quadratic rational functions

TL;DR

This paper introduces a new notion of semistability for non-archimedean rational functions at points in the Berkovich projective line, and analyzes the intrinsic semistability loci for quadratic iterations, revealing stationarity properties similar to polynomial dynamics.

Contribution

It extends potential GIT-semistability to all non-classical points and computes the semistability loci for quadratic iterations using a reduction theoretic slope formula.

Findings

Intrinsic semistability loci are computed for quadratic iterations.

Semistability loci exhibit stationarity similar to polynomial dynamics.

The reduction theoretic slope formula is key to analyzing these loci.

Abstract

We introduce a semistability notion of the intrinsic reductions of a non-archimedean rational function at each non-classical point in the Berkovich projective line, which extends the potential GIT-semistability one defined at each type II points, and compute the intrinsic semistability loci for the iterations of a quadratic rational function using a reduction theoretic slope formula for the hyperbolic resultant function associated to those iterations. In particular, we establish a precise stationarity of those loci for iterated quadratic rational functions similar to that in the case of non-archimedean polynomial dynamics.