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

Reimi Irokawa

TL;DR

This work analyzes meromorphic degenerations of degree $d\ge2$ rational maps on $\mathbb{P}^1(\mathbb{C})$ by passing to the non-archimedean setting over $K=\mathbb{C}((t))$ and employing snc models. The main result shows that the weak limit of the canonical measures $\mu_t$ on the complex fibers is the push-forward $(\mathrm{red}_X)_*\mu_f^{NA}$ of the non-archimedean canonical measure associated to the induced dynamics, providing a unifying framework that generalizes earlier results of DeMarco–Faber and Okuyama to arbitrary snc models. The paper further analyzes contractions to minimal models and provides an illustrative quadratic-polynomial example demonstrating how limiting measures can concentrate on atoms or spread along Berkovich-analytic trees depending on the chosen model. Overall, the approach connects complex degeneration phenomena with non-archimedean dynamics, offering explicit descriptions of weak limits via reduction maps and illustrating how atomic limits emerge from the Berkovich Julia set structure. The results have implications for understanding entropy-maximizing measures under degeneration and for applying non-archimedean techniques to complex dynamical degenerations.

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.

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

TL;DR

This work analyzes meromorphic degenerations of degree rational maps on by passing to the non-archimedean setting over and employing snc models. The main result shows that the weak limit of the canonical measures on the complex fibers is the push-forward of the non-archimedean canonical measure associated to the induced dynamics, providing a unifying framework that generalizes earlier results of DeMarco–Faber and Okuyama to arbitrary snc models. The paper further analyzes contractions to minimal models and provides an illustrative quadratic-polynomial example demonstrating how limiting measures can concentrate on atoms or spread along Berkovich-analytic trees depending on the chosen model. Overall, the approach connects complex degeneration phenomena with non-archimedean dynamics, offering explicit descriptions of weak limits via reduction maps and illustrating how atomic limits emerge from the Berkovich Julia set structure. The results have implications for understanding entropy-maximizing measures under degeneration and for applying non-archimedean techniques to complex dynamical degenerations.

Abstract

For an analytic family 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.

Paper Structure

This paper contains 11 sections, 10 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Berkovich geometry and reduction map
  3. Berkovich analytification of the projective line
  4. Tree structure of Berkovich projective line
  5. Non-archimedean dynamics
  6. Snc models and reduction map
  7. contraction of curves and relation between minimal models
  8. Proof of the main theorem
  9. Example
  10. Non-archimedean dynamics of quadratic polynomials
  11. Weak limits of the canonical measures

Key Result

Proposition 1.1

Let $\{f_t\ |\ t\in\mathbb{D}\}$ be a meromorphic family of rational functions of degree $d\geq2$ that degenerates at $t=0$. The family of measures $\{\mu_t\}_{t\in\mathbb{D}^*}$ converges weakly on $\mathbb{P}^1(\mathbb{C})$ to a countable sum of atomic measures $\mu_0$ as $t\to0$. The measure $\mu

Theorems & Definitions (17)

  • Proposition 1.1: DE-MARCO:2014tn, Theorem B
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 7 more