Intrinsic semistable reduction loci for the iterations of non-archimedean quadratic rational functions

Abstract

We introduce the intrinsic reduction of a non-archimedean rational function at each non-classical point in the Berkovich projective line, which can extend the potential GIT-semistable reduction notion at each type II point to the whole non-classical points reasonably, and compute the intrinsic semistable reduction loci for the iterations of a quadratic rational function using a reduction theoretic slope formula for the hyperbolic resultant function (so for Rumely's resultant one) associated to those iterated quadratic polynomials. In particular, we establish a precise stationarity of the intrinsic semistable reduction loci for iterated quadratic rational functions, which is similar to that in the case of non-archimedean polynomial dynamics.