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

Yûsuke Okuyama

TL;DR

The paper develops intrinsic reductions $\tilde{\phi}_\xi$ and intrinsic depths $\mathrm{depth}_{\vec{v}}\tilde{\phi}_\xi$ on the Berkovich projective line to give conceptual proofs of Rumely's moduli characterization of the type II minimum locus of $\mathrm{ordRes}_\phi$ and to obtain a simplified degenerating limit theorem for maximal entropy measures in complex dynamics. A slope formula for the hyperbolic resultant $\mathrm{hypRes}_\phi$ expresses $d_{\vec{v}}\mathrm{hypRes}_\phi$ in terms of $\mathrm{depth}_{\vec{v}}\tilde{\phi}_\xi$, yielding a reduction-theoretic criterion for the minimum locus. An equidistribution result shows that, for $\xi$ not totally invariant, the normalized sum of intrinsic depths converges to the reduced canonical equilibrium measure $(\pi_\xi)_*\nu_\phi$. Applications include linking to a moduli problem on reductions via GIT stability and providing a simpler degeneration argument for meromorphic complex dynamics, culminating in an upgraded degenerating limit theorem for the family of maximal entropy measures and a purely atomic limiting behavior on $\mathbb{P}^1(\mathbb{C})$.

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 $\operatorname{ordRes}_φ$ on the Berkovich hyperbolic space for a rational function $φ$ 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 $φ$ 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.

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

TL;DR

The paper develops intrinsic reductions and intrinsic depths on the Berkovich projective line to give conceptual proofs of Rumely's moduli characterization of the type II minimum locus of and to obtain a simplified degenerating limit theorem for maximal entropy measures in complex dynamics. A slope formula for the hyperbolic resultant expresses in terms of , yielding a reduction-theoretic criterion for the minimum locus. An equidistribution result shows that, for not totally invariant, the normalized sum of intrinsic depths converges to the reduced canonical equilibrium measure . Applications include linking to a moduli problem on reductions via GIT stability and providing a simpler degeneration argument for meromorphic complex dynamics, culminating in an upgraded degenerating limit theorem for the family of maximal entropy measures and a purely atomic limiting behavior on .

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 on the Berkovich hyperbolic space for a rational function on 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 associated to a meromorphic family of complex rational functions. We introduce the intrinsic reduction of a non-archimedean rational function 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.
Paper Structure (5 sections, 5 theorems, 54 equations)

This paper contains 5 sections, 5 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Reduction theoretic characterization of the minimum locus for the hyperbolic resultant function
  3. Equidistribution property for the intrinsic depths towards the reduced canonical equilibrium measure
  4. Proof of Theorem \ref{['th:slope']} and its application to a moduli problem on reductions
  5. Proof of Theorem \ref{['th:unicity']} and its application to a degeneration problem in complex dynamics

Key Result

Theorem 1

Let $\phi\in K(z)$ be a rational function on $\mathbb{P}^1$ of degree $d>1$. Then for every point $\xi\in\mathsf{H}^1$ and every direction $\vec{v}\in T_\xi\mathsf{P}^1$, in particular, the function $\mathop{\mathrm{hypRes}}\nolimits_\phi$ takes its minimum at $\xi$ (resp. uniquely at $\xi$) if and only if for every $\vec{v}\in T_\xi\mathsf{P}^1$,

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['th:slope']}
  • Theorem 2.1: Rumely17
  • proof : Proof of Theorem \ref{['th:unicity']}
  • Theorem 3: Improved degenerating limit theorem
  • proof : Proof of the first convergence in \ref{['eq:surpluslimit']}
  • Proposition 3.2