Meromorphic degeneration of rational functions over snc models of the projective line

TL;DR

This paper studies how rational functions degenerate analytically over snc models of the projective line, linking their limiting measures to non-archimedean dynamics and generalizing previous results.

Contribution

It introduces a new framework connecting degenerations of rational functions with non-archimedean measures on snc models, extending prior work by DeMarco-Faber and Okuyama.

Findings

Limiting measure is the push forward of the canonical measure on the non-archimedean model.

Generalizes previous results to a broader class of degenerations.

Establishes a link between complex degenerations and non-archimedean dynamics.

Abstract

For an analytic family $\{f_t\}_{t\in\mathbb{D}^*}$ on the unit punctured disk that meromorphically degenerates at the origin, we show that its limiting measure on an snc model is given by the push forward of the canonical measure attached to the non-archimedean rational function naturally induced from the family, which is a generalization of the results by DeMarco-Faber and Okuyama.