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Stable algebraic families of polarized dynamical systems and exceptional extensions

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.AG — Algebraic Geometrymath.CV — Complex Variables

Contributed by Zhuchao Ji

Version 2 / Oct 15, 2026 / CC BY 4.0

Abstract

The notion of exceptional maps in arbitrary dimension was recently introduced by Ji, Song, and Xie. In this paper, we introduce exceptional extensions of polarized dynamical systems: a family of systems admitting a positive-dimensional exceptional factor whose generic fiber dynamics have simultaneous local product coordinates with an isotrivial system. We then prove that an algebraic family of polarized endomorphisms of normal projective varieties over a quasi-projective curve is JJ-stable on a nonempty Zariski open subset if and only if it is isotrivial or it is an exceptional extension after a finite base change. This extends McMullen's theorem for stable algebraic families of rational maps to higher dimension. One ingredient is inspired by Ji and Xie's proof of McMullen's theorem: the use of homoclinic orbits and the multipliers of adjoint periodic points to control local return maps. The new ingredients are a higher-dimensional algebraization method and an inductive rigidity argument for the fiber dynamics, which together yield the exceptional extension structure.

Provenance statement

This manuscript resulted from a series of interactions between a human and GPT6 Astra Ultra. The process began with the human providing the model with the proof for the case of \(\mathbf P^2\). The human’s contribution was to arrive, through these exchanges, at a correct formulation of the statement to be proved, and ask it to focus on tackling the essential difficulties and not to get distracted. With Junyi Xie and Jit-Wu Yap, we hope to write a proper exposition of this in the future.
AI provenance filesStable_families_collaboration_report.md · 59 KB

Tools used

OpenAI
CodexVersion GPT-6 Astra Ultra

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Version history

  1. v1Submitted by Zhuchao JiInitial depositSupersededOct 15, 2026
  2. v2Submitted by Zhuchao JiSubmitted as a new versionCurrentOct 15, 2026