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Semistable intrinsic reduction loci for the iterations of non-archimedean quadratic rational functions

Yûsuke Okuyama

Abstract

We introduce a semistability notion of the intrinsic reductions of a non-archimedean rational function at each non-classical point in the Berkovich projective line, and compute the intrinsic semistability loci for the iterations of a quadratic rational function using a reduction theoretic slope formula for the hyperbolic resultant function associated to those iterations. In particular, we establish a precise stationarity of those loci for iterated quadratic rational functions similar to that in the case of non-archimedean polynomial dynamics.

Semistable intrinsic reduction loci for the iterations of non-archimedean quadratic rational functions

Abstract

We introduce a semistability notion of the intrinsic reductions of a non-archimedean rational function at each non-classical point in the Berkovich projective line, and compute the intrinsic semistability loci for the iterations of a quadratic rational function using a reduction theoretic slope formula for the hyperbolic resultant function associated to those iterations. In particular, we establish a precise stationarity of those loci for iterated quadratic rational functions similar to that in the case of non-archimedean polynomial dynamics.
Paper Structure (6 sections, 1 theorem, 33 equations)

This paper contains 6 sections, 1 theorem, 33 equations.

Table of Contents

  1. Introduction
  2. Intrinsic reductions of a rational function
  3. Main result: the stationarity of semistable intrinsic reduction loci for iterated quadratic rational functions
  4. Background
  5. Proof of Theorem \ref{['th:quad']}
  6. Normal forms in the finite order reduction case: Proof of $\xi_0=\xi_1$

Key Result

Theorem 1

Let $\phi\in K(z)$ be a quadratic rational function on $\mathbb{P}^1$, and let us denote by $\xi_\phi\in\mathsf{H}^1_{\mathrm{II}}$ the unique minimum point of $\mathop{\mathrm{hypRes}}\nolimits_\phi$. Then unless the intrinsic reduction $\tilde{\phi}_{\xi_\phi}$ of $\phi$ at $\xi_\phi$ is of finite where $\xi_1=\xi_1(\phi)$ is a unique boundary point contained in $(\xi_\phi,\xi_0]$ of the Berkovi

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1.1
  • Remark 2.3