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On a degenerating limit theorem of DeMarco--Faber

Yûsuke Okuyama

Abstract

One of our aims is to complement the proof of DeMarco--Faber's degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco--Faber's more conceptual transfer principle between those dynamics.

On a degenerating limit theorem of DeMarco--Faber

Abstract

One of our aims is to complement the proof of DeMarco--Faber's degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco--Faber's more conceptual transfer principle between those dynamics.

Paper Structure

This paper contains 17 sections, 10 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Factorization on $\mathsf{P}^1$ and quantization
  3. The $f$-balanced measures on $\mathsf{P}^1$ and the maximally ramification locus of $f$ in $\mathsf{P}^1$
  4. Main result: the projections of the $f$-balanced measures on $\mathsf{P}^1$ to $\mathsf{P}^1/S(\Gamma_G)$
  5. Application: the degenerating weak limit for the maximal entropy measures on $\mathbb{P}^1(\mathbb{C})$
  6. Background from Berkovich dynamics
  7. Berkovich projective line
  8. Dynamics on $\mathsf{P}^1$ and their reductions
  9. The tangent maps and the directional/surplus local degrees of rational functions
  10. The hyperbolic metric $\rho$ on $\mathsf{H}^1$ and the piecewise affine action of $h$ on $(\mathsf{H}^1,\rho)$
  11. Quantized local degrees and quantized pullbacks
  12. Degenerating balanced property for degenerating weak limit points of the maximal entropy measures on $\mathbb{P}^1(\mathbb{C})$
  13. A direct translation
  14. Proof of Theorem \ref{['th:computation']}
  15. Proof of Theorem \ref{['th:B']}
  16. ...and 2 more sections

Key Result

Theorem A

Let $K$ be an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and non-archimedean absolute value, let $f\in K(z)$ be a rational function on $\mathbb{P}^1$ of degree $d>1$, and suppose that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$ and that $f(a)=a$$ which in particular yields and moreover, the following three statements that $\deg_{f^n(\mathcal{S

Theorems & Definitions (28)

  • Definition 1.1: tame maximal ramification
  • Theorem A
  • Theorem B: DF14
  • Definition 2.9: the quantized local degree
  • Definition 3.3
  • Theorem 3.4
  • Theorem 3.5: a consequence of DF14
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 18 more