Dynamical André-Oort Conjecture

• Dynamical André-Oort Conjecture is a rigidity phenomenon in algebraic families of rational maps that isolates PCF maps as special points within moduli spaces.
• It employs algebraic, Arakelov, and complex dynamical methods to establish criteria and classifications for curves exhibiting Zariski-dense collections of PCF maps.
• The conjecture implies multiplier spectrum rigidity and degree gap phenomena, linking dynamics, geometry, and arithmetic through precise invariants.

The Dynamical André-Oort (DAO) Conjecture formulates and establishes a rigidity phenomenon in algebraic families of rational maps, identifying a precise dynamical analogue to the classical André-Oort conjecture in arithmetic geometry. The conjecture isolates postcritically finite (PCF) maps within the moduli space of rational maps as "special points" and characterizes the algebraic subvarieties that may contain Zariski-dense collections of such maps. Recent advances have established the conjecture for curves, demonstrated robust arithmetic and complex-analytic techniques, and have elucidated rigidity features that connect dynamical, geometric, and arithmetic structures (Xie, 15 Nov 2025, Ji et al., 2023, Ghioca et al., 2015).

1. Moduli Spaces, Special Points, and the Conjecture

Let $\mathrm{Rat}_d(\C)$ denote the affine variety of degree-$d$ rational maps $f:\PP^1\to\PP^1$ over $\C$, and $\cM_d := \mathrm{Rat}_d(\C)/\PGL_2(\C)$ the coarse moduli space, an affine variety of dimension $2d-2$ defined over $\overline{\Q}$ (Xie, 15 Nov 2025). A rational map $f$ is postcritically finite (PCF) if each critical point has finite forward iteration orbit, i.e., its postcritical set

$\PC(f) = \bigcup_{n\ge1} f^n(C(f))$

is finite, where $C(f)$ is the set of critical points.

A Zariski-closed subvariety $V\subset\cM_d$ is called special (in the dynamical sense) if it contains a Zariski-dense set of PCF maps. The DAO conjecture, in its curve case (Baker–DeMarco), posits that for a non-isotrivial algebraic family of maps parameterized by an algebraic curve $V$, the set of PCF fibers is Zariski-dense if and only if the family has at most one independent critical orbit (Xie, 15 Nov 2025, Ji et al., 2023).

2. Statement and Resolution of the DAO Conjecture for Curves

Formal Statement

Let $f:V\times\PP^1\to V\times\PP^1$, $(t,z)\mapsto (t,f_t(z))$, be a non-isotrivial family of degree $d\ge2$ rational maps over a smooth irreducible algebraic curve $V/\C$. Then the following are equivalent:

• The set $\{t\in V(\C): f_t \text{ is PCF}\}$ is infinite/Zariski-dense in $V$.
• The family $f$ has at most one independent critical orbit, i.e., among the $2d-2$ critical points, any two are dynamically related (there is an algebraic curve in $\PP^1\times\PP^1$ containing their orbits, preperiodic under $f\times f$) (Xie, 15 Nov 2025, Ji et al., 2023).

Stronger Bogomolov-Type Theorem

A still stronger result asserts that for any $\varepsilon > 0$, if the "critical height" $h_{\rm crit}(t)$ (sum of Call–Silverman canonical heights of the critical points) satisfies $\#\{ t\in V : h_{\rm crit}(t)<\varepsilon \}$ infinite, then the family must have at most one independent critical orbit (Ji et al., 2023). This yields a dynamical Bogomolov-type statement.

3. Proof Techniques: Algebraic, Arakelov, and Complex Dynamics

The proof of the conjecture for curves synthesizes several strands:

(A) Algebraic Geometry

The construction uses the structure of $\cM_d$ as an affine variety, with PCF points Zariski-dense yet arithmetically rigid (analogous to complex multiplication points in Shimura varieties). Non-isotriviality is crucial: a constant family trivially satisfies the conjecture but does not generate the observed rigidity (Xie, 15 Nov 2025).

(B) Arakelov Theory and Equidistribution

Central is the theory of adelic metrized line bundles and equidistribution of small height points due to Yuan–Zhang. Namely, for a sequence of parameters $t_n$ for which the critical height(s) tend to zero, the associated Galois orbits become equidistributed toward the bifurcation measure $\mu_{f,a}$ on the base $V$, for each active critical point $a$ (Xie, 15 Nov 2025, Ji et al., 2023, Ghioca et al., 2015). Consequently, all bifurcation measures for active critical points must be mutually proportional, which is only possible under strong constraints on orbit independence.

(C) Complex Dynamical and Bifurcation Theory

The pluripotential-theoretic and bifurcation-analytic component analyzes the relative Green current $T_f$ and associated bifurcation measure for marked points: $T_f = \lim_{n\to\infty} d^{-n}(f^n)^*(\omega_{\PP^1}),\quad \mu_{f,a} = a^*(T_f)$ Active critical points (those varying non-trivially in the family) correspond to nonzero bifurcation measures. For typical (in the bifurcation measure sense) parameters, the map $f_t$ is Collet–Eckmann and demonstrates polynomial recurrence, facilitating the construction of renormalization maps that exhibit phase–parameter similarity (Xie, 15 Nov 2025, Ji et al., 2023).

(D) Invariant Correspondences and Rigidity

If two orbits were independent, the phase–parameter similarity would produce nontrivial symmetries of the maximal-entropy measure, which by rigidity theorems can only arise from algebraic correspondences that violate the independence assumption. This contradiction closes the proof (Xie, 15 Nov 2025, Ji et al., 2023).

4. Classification Results: Curves with Infinitely Many Special Points

For specific families, such as unicritical polynomials $z^d+a$, complete classification results are available:

• For curves $C\subset\C^2$, $(a,b)\in C$, with both $z^d+a$ and $z^d+b$ PCF infinitely often, $C$ must be a horizontal/vertical line through a PCF parameter or $b=\zeta a$ for a $(d{-}1)$st root of unity $\zeta$ (Ghioca et al., 2015, Ghioca et al., 2014).
• For cubic polynomials, the classification of special curves corresponds to loci defined by critical orbit relations $f^n(c_i)=f^m(c_j)$ (Ghioca et al., 2016).

These results parallel André-Oort-type theorems for Shimura or modular curves, where special subvarieties are characterized by functional relations among moduli parameters.

5. Multiplier Spectrum Rigidity and Degree Gap Phenomena

The DAO conjecture supports profound rigidity applications:

• Multiplier Spectrum Map: The (truncated) multiplier spectrum map assigns finite sets of multipliers of periodic points, up to conjugacy, and is generically injective in the moduli space $\cM_d$. Thus, for generic $[f]\in\cM_d$, the multiplier spectrum uniquely determines the conjugacy class (Xie, 15 Nov 2025).
• Degree Gaps: In non-isotrivial families, there can be no non-zero marked multiplier with degree strictly less than the global critical height, tightly restricting possible behaviors in families with infinitely many PCF specializations (Ingram, 2020).

6. Key Lemmas, Intermediate Results, and Representative Examples

Lemma: The vanishing of the bifurcation measure $\mu_{f,a}$ at a marked point characterizes persistent preperiodicity: $$\mu_{f,a} = 0 \iff a(t)\text{ is persistently preperiodic for all }f_t$$ Equidistribution theorems assert

$\frac{1}{|\Gal(t_n)|}\sum_{\sigma\in\Gal(t_n)}\delta_\sigma \to \frac{\mu_{f,a}}{\mu_{f,a}(V)}$

for any sequence of parameters with vanishing canonical heights (Xie, 15 Nov 2025, Ji et al., 2023).

Examples in degree $d$ unicritical families show that only the horizontal/vertical/diagonal loci (with multiplicative constants being roots of unity) can contain infinitely many simultaneous PCF pairs; all such classification results are proved via analysis combining equidistribution, bifurcation geometry, and functional equations induced by Green function proportionality (Ghioca et al., 2015, Ghioca et al., 2014).

7. Impact, Applications, and Perspectives

The DAO conjecture, now a theorem for curves, establishes dynamical analogues of deep results in arithmetic geometry and provides a bridge between unlikely intersection theory, arithmetic dynamics, and rigidity in complex dynamics. It enables generic injectivity results for parameter invariants such as multiplier spectra, and constrains the geometry of special subvarieties in moduli spaces.

While the extension to higher-dimensional parameter spaces remains open, the methodologies developed—synthesizing adelic, pluripotential, and combinatorial rigidity—are anticipated to extend to broader dynamical and arithmetic settings, including higher-degree and higher-dimensional cases (Xie, 15 Nov 2025, Ji et al., 2023, Ingram, 2020).

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Key References:

• "Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture" (Xie, 15 Nov 2025)
• "DAO for curves" (Ji et al., 2023)
• "The Dynamical Andre-Oort Conjecture: Unicritical Polynomials" (Ghioca et al., 2015)
• "Degree gaps for multipliers and the dynamical Andre-Oort conjecture" (Ingram, 2020)
• "The Dynamical Andre-Oort Conjecture for cubic polynomials" (Ghioca et al., 2016)
• "A case of the Dynamical Andre-Oort Conjecture" (Ghioca et al., 2014)