The intrinsic reductions and the intrinsic depths in non-archimedean dynamics

Abstract

In this short paper, we aim at giving a more conceptual and simpler proof of Rumely's moduli theoretic characterization of type II minimal locus of the resultant function $\operatorname{ordRes}_\phi$ on the Berkovich hyperbolic space for a rational function $\phi$ on $\mathbb{P}^1$ defined over an algebraically closed and complete field that is equipped with a non-trivial and non-archimedean absolute value, and also aim at giving a much simpler and more natural proof of a degenerating limit theorem, in an improved form after DeMarco--Faber, for the family of the unique maximal entropy measures on $\mathbb{P}^1(\mathbb{C})$ associated to a meromorphic family of complex rational functions. We introduce the intrinsic reduction of a non-archimedean rational function $\phi$ at each point in the Berkovich projective line and its directionwise intrinsic depths, which are suitable notions for the above aims and defined in terms of the tree and analytic structures of the Berkovich projective line. Then we establish two theorems in non-archimedean dynamics, both of which play key roles in the above aims.