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Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach

Thomas Gauthier, Johan Taflin, Gabriel Vigny

TL;DR

The paper develops a bridge between arithmetic dynamics and complex dynamics to address sparsity and uniformity of postcritically finite maps on projective spaces. It introduces higher-order dynamical currents and dynamical volumes for families of subvarieties, proving that PCF maps cannot be Zariski-dense in the moduli spaces $\mathscr{M}_d^k$ and $\mathscr{P}_d^2$ by showing that their would-be equidistribution would contradict the nontrivial bifurcation measure. A key technical advance is the construction of open sets in End$_d^k$ where bifurcation phenomena are maximal via blender mechanisms, enabling positive-volume current arguments and height inequalities. The authors also prove a uniform bound on the number of preperiodic points in the critical set for regular polynomial endomorphisms of $\mathbb{A}^2$, and derive dynamical relative and parametric equidistribution results that connect moduli heights with dynamical heights. The results illuminate the deep interplay between complex-analytic bifurcation structures and arithmetic heights, providing new rigidity, uniformity, and sparsity conclusions in higher-dimensional dynamics.

Abstract

An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that $f^m(\mathrm{Crit}(f))\subseteq f^n(\mathrm{Crit}(f))$. When $k\geq2$, it was conjectured by Ingram, Ramadas and Silverman that, in the space $\mathrm{End}_d^k$ of all endomorphisms of degree $d$ of $\mathbb{P}^k$, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space $\mathrm{Poly}_d^2$ of all regular polynomial endomorphisms of degree $d\geq2$ of the affine plane $\mathbb{A}^2$, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.

Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach

TL;DR

The paper develops a bridge between arithmetic dynamics and complex dynamics to address sparsity and uniformity of postcritically finite maps on projective spaces. It introduces higher-order dynamical currents and dynamical volumes for families of subvarieties, proving that PCF maps cannot be Zariski-dense in the moduli spaces and by showing that their would-be equidistribution would contradict the nontrivial bifurcation measure. A key technical advance is the construction of open sets in End where bifurcation phenomena are maximal via blender mechanisms, enabling positive-volume current arguments and height inequalities. The authors also prove a uniform bound on the number of preperiodic points in the critical set for regular polynomial endomorphisms of , and derive dynamical relative and parametric equidistribution results that connect moduli heights with dynamical heights. The results illuminate the deep interplay between complex-analytic bifurcation structures and arithmetic heights, providing new rigidity, uniformity, and sparsity conclusions in higher-dimensional dynamics.

Abstract

An endomorphism of degree is said to be postcritically finite (or PCF) if its critical set is preperiodic, i.e. if there are integers such that . When , it was conjectured by Ingram, Ramadas and Silverman that, in the space of all endomorphisms of degree of , such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space of all regular polynomial endomorphisms of degree of the affine plane , we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.
Paper Structure (41 sections, 280 equations, 2 figures)

This paper contains 41 sections, 280 equations, 2 figures.

Table of Contents

  1. The dynamical volumes of a family of subvarieties
  2. The canonical height over a function field of characteristic zero
  3. The higher bifurcation currents of a pair
  4. The dynamical volumes of a pair
  5. Dynamical volume as limits of iterated intersection numbers
  6. A sufficient criterion for positive volume
  7. Rigidity of some stable families
  8. Spaces of endomorphisms, moduli spaces, stable families
  9. The spaces $\mathrm{End}_d^k$ and $\mathrm{Poly}_d^k$
  10. The moduli spaces $\mathscr{M}_d^k$ and $\mathscr{P}_d^k$ and good families
  11. Stable families of endomorphisms of $\mathbb{P}^k$ following Berteloot-Bianchi-Dupont
  12. Families with many constant multipliers
  13. Families of regular polynomial endomorphisms of the affine plane
  14. Blenders and the bifurcation measures
  15. Notations
  16. ...and 26 more sections

Figures (2)

  • Figure 1: Summary of the notations. The whole picture is contained in $\mathcal{U}_-.$ In the example obtained in Section \ref{['sec-existence']}, the hyperbolic set $\Lambda(f)$ is a Cantor set but it intersects any sufficiently vertical graph in $\mathcal{V}_-.$
  • Figure 2: Definition of $\Gamma_{j,l}(f)$ where $x(f)\in W^u_{p(f),loc}$ is a preimage of $r(f).$ The integers $m$ and $a$ are constant but $j$ and $l$ can be large. The next two lemmas show that $c_j(f)$ and $c_{j,l}(f)$ are essentially equal to $\chi_{p}(f)^j$ and $\chi_p(f)^j\chi_r(f)^l$ respectively in the coordinates on $W^{cu}_{r(f),loc}$ given by $\phi_f.$

Theorems & Definitions (56)

  • proof
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  • proof : Proof of Proposition \ref{['tm:densitypart1']}
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  • ...and 46 more