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Uniform bound on common periodic points for families of regular plane polynomial automorphisms

Marc Abboud, Yugang Zhang

TL;DR

The paper proves a uniform bound on the number of common periodic points for one‑dimensional arithmetic families of dissipative regular plane polynomial automorphisms, unless the two families share a common iterate. It builds an adelic and geometric height framework using boundary topology, invariant adelic divisors, and Green functions, and leverages Yuan–Zhang equidistribution to translate small-height phenomena into dynamical rigidity. The authors show that either a common iterate exists for the fibers, or the intersection of the fiberwise periodic point sets is uniformly bounded by a constant depending only on the families, with precise distinctions dictated by isotriviality and Jacobian behavior. This advances uniformity results from rational maps to plane polynomial automorphisms through a synthesis of arithmetic dynamics, equidistribution, and pluripotential theory, highlighting the role of dissipativity and fiberwise invariants in controlling global dynamics.

Abstract

Given two one-dimensional families $f$ and $g$ of regular plane polynomial automorphisms parameterised by an algebraic curve $B$, all defined over some number field $K$, such that one of them is dissipative, we prove that at any parameter $b\in B(\mathbb{C})$, either $f_b$ and $g_b$ share a common iterate, or the number of their common periodic points $\mathrm{Per}(f_b) \cap \mathrm{Per}(g_b)$ is bounded by a uniform constant $D$ (independent of the parameter $b$). We thus extend a result of Mavraki and Schmidt for rational maps to our setting.

Uniform bound on common periodic points for families of regular plane polynomial automorphisms

TL;DR

The paper proves a uniform bound on the number of common periodic points for one‑dimensional arithmetic families of dissipative regular plane polynomial automorphisms, unless the two families share a common iterate. It builds an adelic and geometric height framework using boundary topology, invariant adelic divisors, and Green functions, and leverages Yuan–Zhang equidistribution to translate small-height phenomena into dynamical rigidity. The authors show that either a common iterate exists for the fibers, or the intersection of the fiberwise periodic point sets is uniformly bounded by a constant depending only on the families, with precise distinctions dictated by isotriviality and Jacobian behavior. This advances uniformity results from rational maps to plane polynomial automorphisms through a synthesis of arithmetic dynamics, equidistribution, and pluripotential theory, highlighting the role of dissipativity and fiberwise invariants in controlling global dynamics.

Abstract

Given two one-dimensional families and of regular plane polynomial automorphisms parameterised by an algebraic curve , all defined over some number field , such that one of them is dissipative, we prove that at any parameter , either and share a common iterate, or the number of their common periodic points is bounded by a uniform constant (independent of the parameter ). We thus extend a result of Mavraki and Schmidt for rational maps to our setting.
Paper Structure (17 sections, 22 theorems, 73 equations)

This paper contains 17 sections, 22 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background and the strategy of Main theorems
  4. Adelic divisors and line bundles over quasiprojective varieties
  5. $b$-divisors
  6. Over a metrised field
  7. Over a projective variety
  8. Over a quasiprojective variety
  9. In the geometric setting
  10. Restriction of adelic divisors and line bundles
  11. Invariant adelic divisor
  12. Green functions and equilibrium measure over a complete metrised field
  13. Over a number field
  14. Over a function field
  15. In a family
  16. ...and 2 more sections

Key Result

Theorem A

Let $f$ and $g$ be two arithmetic families of regular plane polynomial automorphisms and suppose $f$ is dissipative. Then there exist a positive constant $D> 0$ depending only on $f$ and $g$, and two positive integers $N,M>0$, such that for any $b \in B ( \mathbf{C})$, either or And the equality eq:equalitynm can appear for at most finitely many parameters unless $f^N = g^M$ globally.

Theorems & Definitions (35)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Theorem 2.1: yuanAdelicLineBundles2026guoIntegrationFormulaChern2025
  • Proposition 2.2
  • Theorem 2.3: yuanAdelicLineBundles2026
  • proof
  • Proposition 2.4
  • Corollary 2.5
  • proof
  • ...and 25 more