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Unlikely intersections problem for automorphisms of Markov surfaces

Marc Abboud

TL;DR

This work analyzes unlikely intersections for automorphisms of Markov surfaces \\mathcal{M}_D with positive entropy, proving that for special parameters D=0 or D=2-2\\cos(\\pi/q) the condition that two loxodromic automorphisms share a Zariski-dense set of periodic points is equivalent to the existence of common iterates. The authors develop Green-function calculus and adelic divisor techniques across archimedean and non-archimedean places, and combine this with representation-theoretic descriptions (including quasi-Fuchsian theory) to link dynamics to equilibrium measures. They establish invariant adelic divisors, prove equidistribution of periodic points via Yuan-Zhang, and derive rigidity statements by showing saddle points must lie in the equilibrium support; the main result shows that, except for D=4, two loxodromic automorphisms sharing a dense set of periodic points must share an iterate. The results illuminate the interplay between arithmetic, complex dynamics, and hyperbolic geometry on affine surfaces, offering a framework for analyzing dynamical rigidity in non-projective settings and near-parabolic parameter regimes. The transcendental-parameter case is also handled, proving the equivalence of equality of periodic points and common iterates in that regime, underscoring the robustness of the rigidity phenomenon beyond algebraic parameters.

Abstract

We study the problem of unlikely intersections for automorphisms of Markov surfaces of positive entropy. We show for certain parameters that two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate. Our proof uses arithmetic equidistribution for adelic line bundles over quasiprojective varieties, the theory of laminar currents and quasi-Fuchsian representation theory.

Unlikely intersections problem for automorphisms of Markov surfaces

TL;DR

This work analyzes unlikely intersections for automorphisms of Markov surfaces \\mathcal{M}_D with positive entropy, proving that for special parameters D=0 or D=2-2\\cos(\\pi/q) the condition that two loxodromic automorphisms share a Zariski-dense set of periodic points is equivalent to the existence of common iterates. The authors develop Green-function calculus and adelic divisor techniques across archimedean and non-archimedean places, and combine this with representation-theoretic descriptions (including quasi-Fuchsian theory) to link dynamics to equilibrium measures. They establish invariant adelic divisors, prove equidistribution of periodic points via Yuan-Zhang, and derive rigidity statements by showing saddle points must lie in the equilibrium support; the main result shows that, except for D=4, two loxodromic automorphisms sharing a dense set of periodic points must share an iterate. The results illuminate the interplay between arithmetic, complex dynamics, and hyperbolic geometry on affine surfaces, offering a framework for analyzing dynamical rigidity in non-projective settings and near-parabolic parameter regimes. The transcendental-parameter case is also handled, proving the equivalence of equality of periodic points and common iterates in that regime, underscoring the robustness of the rigidity phenomenon beyond algebraic parameters.

Abstract

We study the problem of unlikely intersections for automorphisms of Markov surfaces of positive entropy. We show for certain parameters that two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate. Our proof uses arithmetic equidistribution for adelic line bundles over quasiprojective varieties, the theory of laminar currents and quasi-Fuchsian representation theory.
Paper Structure (47 sections, 49 theorems, 160 equations)

This paper contains 47 sections, 49 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Unlikely intersections problem for special parameters
  3. Green functions
  4. Arithmetic equidistribution
  5. The Picard parameter $D = 4$
  6. For a general parameter
  7. Plan of the paper
  8. Adelic divisors over quasiprojective varieties
  9. Weil and Cartier divisors
  10. Berkovich spaces
  11. Places and restricted analytic spaces
  12. Adelic divisors over a projective variety
  13. Over quasiprojective varieties
  14. Picard-Manin space at infinity
  15. Compactifications
  16. ...and 32 more sections

Key Result

Theorem A

Let $D = 0$ or $D = 2 - 2 \cos (\frac{\pi}{q})$ with $q \in \mathbf{Z}_{\geq 2}$. If $f$ and $g$ are two loxodromic automorphisms of $\cM_D$, then the following statements are equivalent.

Theorems & Definitions (92)

  • Theorem A
  • Theorem B
  • Proposition 2.1: mustataWeightFunctionsNonArchimedean2015
  • Corollary 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5: Lemma 3.3.3 of yuanAdelicLineBundles2023
  • proof
  • Corollary 2.6
  • ...and 82 more