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

Thomas Gauthier, Johan Taflin+1 more|arXiv (Cornell University)|May 3, 2023

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper proves that postcritically finite (PCF) maps in the space of degree-$d$ endomorphisms of $\mathbb{P}^k$ ($k \geq 2$) are not Zariski dense, using a complex analytic approach combining dynamical heights, bifurcation currents, and equidistribution. It further establishes a uniform bound on the number of preperiodic points in the critical set for a Zariski open subset of regular polynomial endomorphisms in $\mathbb{A}^2$, leveraging arithmetic dynamics and real dynamics techniques.

ABSTRACT

An endomorphism $f:\mathbb{P}^k o\mathbb{P}^k$ 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.

Motivation & Objective

• To resolve a conjecture in complex dynamics stating that postcritically finite (PCF) endomorphisms of $\mathbb{P}^k$ ($k \geq 2$) are not Zariski dense in the moduli space $\mathscr{M}_d^k$.
• To establish a uniform bound on the number of preperiodic points in the critical set for a Zariski open subset of regular polynomial endomorphisms in $\mathbb{A}^2$.
• To link dynamical stability via bifurcation measures with arithmetic dynamical heights, using height inequalities and equidistribution theorems.
• To extend results on uniform bounds in arithmetic dynamics to higher-dimensional dynamical systems using moduli-theoretic and analytic techniques.

Proposed method

• Use of canonical heights on subvarieties to measure arithmetic-dynamical complexity, particularly for the critical set in families of endomorphisms.
• Application of the theory of dynamical heights and DSH functions to relate canonical heights to the bifurcation current and measure of instability.
• Construction of a Zariski open subset in $\mathrm{End}_d^k$ where the support of the bifurcation measure has non-empty interior, using methods from real dynamics and blenders.
• Employment of equidistribution results for preperiodic points with respect to the canonical measure to control the number of such points in the critical set.
• Reduction of the problem to a moduli space setting via the universal family over $\mathscr{M}_d^k$ and $\mathscr{P}_d^2$, leveraging automorphism invariance of heights.
• Use of uniform bounds from arithmetic dynamics, inspired by DeMarco-Krieger-Ye and Dimitrov-Gao-Habegger, to establish finiteness in the critical set under height constraints.

Experimental results

Research questions

• RQ1 Are postcritically finite endomorphisms of $\mathbb{P}^k$ ($k \geq 2$) Zariski dense in the moduli space $\mathscr{M}_d^k$?
• RQ2 Can a uniform bound be established on the number of preperiodic points in the critical set of regular polynomial endomorphisms in $\mathbb{A}^2$?
• RQ3 Does the support of the bifurcation measure have non-empty interior in the moduli space of regular polynomial endomorphisms in $\mathbb{A}^2$?
• RQ4 How can the canonical height of the critical set be used to control the number of rational preperiodic points in the critical set?
• RQ5 To what extent does the correspondence between an endomorphism and the multipliers of its periodic points fail to be finite-to-one?

Key findings

• Postcritically finite endomorphisms of $\mathbb{P}^k$ ($k \geq 2$) are not Zariski dense in $\mathscr{M}_d^k$, confirming a conjecture in the literature.
• There exists a non-empty Zariski open subset $U \subset \mathrm{Poly}_d^2$ such that for all $f \in U(\bar{\mathbb{Q}})$, the number of $\bar{\mathbb{Q}}$-rational preperiodic points in the critical set is bounded by a constant $B(d)$ depending only on $d$.
• For any $f \in U(\mathbb{C})$, the number of complex preperiodic points in the critical set is bounded by $B(p,d)$, a constant depending on the base polynomial $p$ and degree $d$.
• The support of the bifurcation measure has non-empty interior in both $\mathscr{M}_d^k$ and $\mathscr{P}_d^2$, implying maximal instability in these families.
• The correspondence between a regular polynomial endomorphism in $\mathscr{P}_d^2$ and the multipliers of its periodic points is finite-to-one outside a Zariski closed set.
• A uniform bound $\varepsilon(p,d) > 0$ and $B(p,d) \geq 1$ exist such that $\#\{z \in C_f(\bar{\mathbb{Q}}) : \widehat{h}_f(z) \leq \varepsilon(p,d)\} \leq B(p,d)$ for all $f$ in a Zariski open subset of $\mathrm{Sk}(p,d)$.