Relative Dynamical Manin–Mumford Conjecture

• Relative Dynamical Manin–Mumford Conjecture is a framework in arithmetic and complex dynamics that uses invariant Green currents to characterize subvarieties with Zariski-dense preperiodic points.
• It unifies classical Manin–Mumford with fixed-map and toric dynamics via a family-theoretic approach, equating current nonvanishing with the presence of dense dynamically special points.
• The conjecture employs techniques from complex analysis, hyperbolic dynamics, and Frobenius lifts, while posing open challenges in higher dimensions and non-abelian settings.

Relative Dynamical Manin–Mumford Conjecture denotes a class of problems in arithmetic and complex dynamics that seek a geometric characterization of subvarieties carrying a Zariski-dense set of dynamically special points. In its modern family-theoretic form, the conjecture concerns an algebraic family of endomorphisms $\Phi:S\times \mathbb{P}^N\to S\times \mathbb{P}^N$, $\Phi(s,z)=(s,f_s(z))$, and predicts that an irreducible subvariety $X\subset S\times \mathbb{P}^N$ flat over $S$ contains a Zariski-dense set of $\Phi$-preperiodic points if and only if a distinguished wedge power of the relative Green current does not vanish on $X$ (DeMarco et al., 2024). Closely related relative statements appear in toric dynamics, affine polynomial automorphisms, polynomial dynamics on $\mathbb{P}^1\times \mathbb{P}^1$, and $p$-adic Frobenius-lift settings, where “specialness” is expressed respectively through quotient tori, reversibility, diagonal-preimage relations, or periodicity under Frobenius lifts (Lin, 2017, Dujardin et al., 2014, Schmidt, 2020, Xie, 2016).

1. Family-theoretic conjectural framework

The formulation developed for families on $S\times \mathbb{P}^N$ begins with the fiberwise preperiodic locus

$$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$

where $\Phi(s,z)=(s,f_s(z))$0 is preperiodic if $\Phi(s,z)=(s,f_s(z))$1 for some $\Phi(s,z)=(s,f_s(z))$2. The family carries a relative invariant Green current

$\Phi(s,z)=(s,f_s(z))$3

with $\Phi(s,z)=(s,f_s(z))$4. Fiberwise, the slice of $\Phi(s,z)=(s,f_s(z))$5 at $\Phi(s,z)=(s,f_s(z))$6 is the usual Green current $\Phi(s,z)=(s,f_s(z))$7 of the map $\Phi(s,z)=(s,f_s(z))$8 (DeMarco et al., 2024).

The conjecture is not stated merely in terms of preperiodicity of $\Phi(s,z)=(s,f_s(z))$9 itself. Instead it uses the notion of a $X\subset S\times \mathbb{P}^N$0-special subvariety. An irreducible $X\subset S\times \mathbb{P}^N$1 is $X\subset S\times \mathbb{P}^N$2-special if, over the function field $X\subset S\times \mathbb{P}^N$3, it is contained in a subvariety $X\subset S\times \mathbb{P}^N$4 carrying a polarizable endomorphism $X\subset S\times \mathbb{P}^N$5 that commutes with an iterate of $X\subset S\times \mathbb{P}^N$6, and $X\subset S\times \mathbb{P}^N$7 is preperiodic under $X\subset S\times \mathbb{P}^N$8. The associated relative special dimension is

$X\subset S\times \mathbb{P}^N$9

where $S$0 is the generic fiber dimension (DeMarco et al., 2024).

The Relative Dynamical Manin–Mumford Conjecture, in the sense used by DeMarco–Mavraki and by subsequent work, asserts the equivalence

$S$1

and

$S$2

Equivalently, if one defines

$S$3

then the conjecture becomes $S$4 (DeMarco et al., 2024). A structural subtlety, emphasized in the formulation itself, is that intersections of $S$5-special subvarieties need not be $S$6-special, so there need not be a unique minimal special subvariety containing $S$7.

2. Relation to classical Manin–Mumford and fixed-map dynamical Manin–Mumford

The conjecture is designed to recover classical and fixed-map statements as special cases. When $S$8 is a point, it reduces to the Dynamical Manin–Mumford problem for a single polarized endomorphism of $S$9. In that setting, $\Phi$0 is the minimal dimension of a special subvariety containing $\Phi$1, and the current-theoretic condition becomes the statement that $\Phi$2 itself is special. Thus the family-theoretic formulation specializes to Zhang’s dynamical Manin–Mumford problem for fixed maps (DeMarco et al., 2024).

For abelian families, the same formalism recovers Relative Manin–Mumford. If $\Phi$3 is induced by multiplication $\Phi$4 on an abelian scheme $\Phi$5, the restriction of $\Phi$6 is the Betti form $\Phi$7, satisfying $\Phi$8. In this case $\Phi$9, and $X$0 equals the relative dimension $X$1 of the abelian scheme, so the condition $X$2 is exactly the maximal Betti-rank condition appearing in Gao–Habegger’s relative Manin–Mumford theorem (DeMarco et al., 2024).

At the opposite extreme, generic endomorphisms of $X$3 exhibit maximal rigidity. For a generic endomorphism $X$4 of degree $X$5, there are no positive-dimensional proper preperiodic subvarieties, any infinite set of preperiodic points is Zariski dense in $X$6, and any infinite subset of a single orbit is Zariski dense as well (Fakhruddin, 2012). In such a regime, any relative statement over a family whose very general fibers are generic collapses to the conclusion that a fiber containing a Zariski-dense set of preperiodic points must be the whole fiber.

3. Established implications and verified model cases

The most general theorem presently available in the family-theoretic framework is the implication from Green-current nonvanishing to density of preperiodic points. If $X$7 is irreducible and flat over $X$8, and

$X$9

then $\mathbb{P}^1\times \mathbb{P}^1$0 is Zariski dense in $\mathbb{P}^1\times \mathbb{P}^1$1 (DeMarco et al., 2024). The proof passes through a $\mathbb{P}^1\times \mathbb{P}^1$2-special container $\mathbb{P}^1\times \mathbb{P}^1$3 of relative dimension $\mathbb{P}^1\times \mathbb{P}^1$4, identifies $\mathbb{P}^1\times \mathbb{P}^1$5 with the invariant current of a commuting polarizable map, uses fiberwise measures of maximal entropy, and then applies hyperbolic sets, holomorphic motions, uniformly laminar currents, and Dujardin’s forced-intersection method to produce Zariski-dense repelling cycles and hence preperiodic points (DeMarco et al., 2024).

The converse direction is known in full for $\mathbb{P}^1\times \mathbb{P}^1$6 and for abelian families, but remains open in higher dimension $\mathbb{P}^1\times \mathbb{P}^1$7 (DeMarco et al., 2024). In dimension one, the conjecture is identified with the dichotomy between persistent preperiodicity and dynamical instability of a marked point; in the abelian setting it is equivalent to maximal Betti rank (DeMarco et al., 2024).

A fixed-map verification in dimension two is provided for regular polynomial endomorphisms on $\mathbb{P}^1\times \mathbb{P}^1$8. If $\mathbb{P}^1\times \mathbb{P}^1$9 is a regular polynomial endomorphism of degree $p$0 and $p$1 is a subvariety of the moduli space of effective divisors of degree $p$2 containing a Zariski-dense set of periodic curves under $p$3, then, after replacing $p$4 by an iterate, a generic curve $p$5 with $p$6 satisfies $p$7, and $p$8 is invariant under the induced endomorphism $p$9 on the divisor moduli space (Zhong, 19 Aug 2025). This is explicitly described as a Dynamical Manin–Mumford type statement on the moduli space of divisors and as a weak verification of the relative conjecture for constant families in the case $S\times \mathbb{P}^N$0 (Zhong, 19 Aug 2025).

Another verified model case occurs in the $S\times \mathbb{P}^N$1-adic setting of lifts of Frobenius. For a lift of Frobenius $S\times \mathbb{P}^N$2 over $S\times \mathbb{P}^N$3, if an irreducible subvariety $S\times \mathbb{P}^N$4 contains a Zariski-dense set of periodic points, then $S\times \mathbb{P}^N$5 is periodic (Xie, 2016). An embedding theorem then transfers this statement to polarized lifts of Frobenius on general projective varieties, producing a relative-over-$S\times \mathbb{P}^N$6 prototype in which “many periodic points” again force periodicity (Xie, 2016).

4. Toric and monomial relative formulations

A particularly explicit relative theorem is available for monomial maps on algebraic tori and toric varieties. For an integer matrix $S\times \mathbb{P}^N$7, the associated monomial endomorphism on the torus $S\times \mathbb{P}^N$8 is

$S\times \mathbb{P}^N$9

On a toric variety $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$0, it extends to a $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$1-equivariant rational self-map $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$2, with well-defined locus

$$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$3

and orbit decomposition

$$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$4

determined by the orbit–cone correspondence (Lin, 2017).

On each torus orbit $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$5, the restriction is again monomial, and the preperiodic points take a uniform algebraic form: $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$6 where $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$7 is a connected algebraic subgroup and $$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$8 is its divisible hull (Lin, 2017). This yields a stratumwise quotient map

$$\operatorname{PrePer}(\Phi)=\{(s,x)\in S\times \mathbb{P}^N : x \text{ is preperiodic for } f_s\},$$9

that transforms preperiodic points into torsion points in the quotient torus.

The resulting relative Dynamical Manin–Mumford statement is orbitwise. If $\Phi(s,z)=(s,f_s(z))$00 is a closed irreducible subvariety and $\Phi(s,z)=(s,f_s(z))$01 is the unique cone such that $\Phi(s,z)=(s,f_s(z))$02 is Zariski dense in $\Phi(s,z)=(s,f_s(z))$03, then preperiodic points are Zariski dense in $\Phi(s,z)=(s,f_s(z))$04 if and only if

$\Phi(s,z)=(s,f_s(z))$05

is a torsion translate of an algebraic subgroup of $\Phi(s,z)=(s,f_s(z))$06 (Lin, 2017). When $\Phi(s,z)=(s,f_s(z))$07, this reduces to Laurent’s theorem for torsion points on tori. The theorem is “relative” because the correct specialness condition is not imposed on $\Phi(s,z)=(s,f_s(z))$08 itself but only after quotienting by the subgroup that captures the intrinsic preperiodic structure of the relevant toric stratum.

This toric theory is closely tied to arithmetic invariants. For monomial maps, the dynamical degrees satisfy $\Phi(s,z)=(s,f_s(z))$09, and the arithmetic degrees on toric varieties are exactly $\Phi(s,z)=(s,f_s(z))$10, where the $\Phi(s,z)=(s,f_s(z))$11 are the irreducible factors of $\Phi(s,z)=(s,f_s(z))$12 over $\Phi(s,z)=(s,f_s(z))$13 (Lin, 2017). These height-growth results support the orbitwise classification but do not yield a general canonical-height characterization of preperiodicity, since the normalized canonical-height sequence may have infinitely many accumulation points (Lin, 2017).

5. Non-abelian relative phenomena

In non-abelian affine dynamics, the relative Dynamical Manin–Mumford problem can require a different notion of specialness. For a Hénon-type polynomial automorphism $\Phi(s,z)=(s,f_s(z))$14 of $\Phi(s,z)=(s,f_s(z))$15, Dujardin and Favre formulate a relative problem for curves $\Phi(s,z)=(s,f_s(z))$16 containing infinitely many periodic points. Their conjecture is that this occurs if and only if there exists an involution $\Phi(s,z)=(s,f_s(z))$17 with $\Phi(s,z)=(s,f_s(z))$18 and some $\Phi(s,z)=(s,f_s(z))$19 such that

$\Phi(s,z)=(s,f_s(z))$20

Thus specialness is expressed through time-reversal symmetry rather than preperiodicity of the curve itself (Dujardin et al., 2014).

The motivation for this shift is geometric: Bedford–Smillie proved that a Hénon-type automorphism admits no invariant algebraic curve, so the classical conclusion “$\Phi(s,z)=(s,f_s(z))$21 is preperiodic” is unavailable in the affine plane (Dujardin et al., 2014). What is proved are strong necessary conditions. If a curve contains infinitely many periodic points, then $\Phi(s,z)=(s,f_s(z))$22 is algebraic and every Galois conjugate has modulus $\Phi(s,z)=(s,f_s(z))$23; under a transversality hypothesis, $\Phi(s,z)=(s,f_s(z))$24 is a root of unity (Dujardin et al., 2014). Conversely, if there is an archimedean place $\Phi(s,z)=(s,f_s(z))$25 with $\Phi(s,z)=(s,f_s(z))$26, then no curve contains infinitely many periodic points (Dujardin et al., 2014). In this setting, the relative DMM phenomenon is therefore mediated by reversibility and Jacobian rigidity.

A different relative variant appears for polynomial dynamics on $\Phi(s,z)=(s,f_s(z))$27, where one studies pairs of points belonging to the same small orbit or grand orbit of a base point $\Phi(s,z)=(s,f_s(z))$28. For a non-exceptional polynomial $\Phi(s,z)=(s,f_s(z))$29 of degree $\Phi(s,z)=(s,f_s(z))$30 and a non-preperiodic $\Phi(s,z)=(s,f_s(z))$31, if a geometrically irreducible non-fibral curve $\Phi(s,z)=(s,f_s(z))$32 satisfies $\Phi(s,z)=(s,f_s(z))$33 infinite, then

$\Phi(s,z)=(s,f_s(z))$34

for some $\Phi(s,z)=(s,f_s(z))$35, where $\Phi(s,z)=(s,f_s(z))$36 is the diagonal (Schmidt, 2020). Under a $\Phi(s,z)=(s,f_s(z))$37-adic escape condition and good reduction coprime to $\Phi(s,z)=(s,f_s(z))$38, the analogous grand-orbit statement becomes

$\Phi(s,z)=(s,f_s(z))$39

for some $\Phi(s,z)=(s,f_s(z))$40 (Schmidt, 2020). Here the relative special subvarieties are precisely fibral curves and iterated preimages of the diagonal; for exceptional maps one must enlarge the classification to “stratified” curves coming from the underlying algebraic-group structure (Schmidt, 2020).

6. Techniques, limitations, and open directions

The subject is methodologically heterogeneous. In the family-theoretic formulation on $\Phi(s,z)=(s,f_s(z))$41, the central tools are Bedford–Taylor products, invariant Green currents, measures of maximal entropy, hyperbolic sets, holomorphic motions, and laminar-current intersection theory (DeMarco et al., 2024). In the toric setting, the key inputs are orbit–cone correspondence, lattice quotients, spectral radii of exterior powers, and Laurent’s theorem on torsion translates (Lin, 2017). In affine plane problems, equidistribution of small points, Green functions $\Phi(s,z)=(s,f_s(z))$42, Pesin theory, and non-archimedean renormalization dominate (Dujardin et al., 2014). For Frobenius lifts, perfectoid inverse limits, tilting, and characteristic-$\Phi(s,z)=(s,f_s(z))$43 approximation replace complex-analytic methods (Xie, 2016).

Several limitations are explicit. The general conjectural equivalence

$\Phi(s,z)=(s,f_s(z))$44

is proved only in the direction from current nonvanishing to density; the converse remains open for $\Phi(s,z)=(s,f_s(z))$45 outside known cases such as $\Phi(s,z)=(s,f_s(z))$46 and abelian families (DeMarco et al., 2024). The classification of $\Phi(s,z)=(s,f_s(z))$47-special subvarieties is intrinsically delicate because intersections of special loci need not remain special (DeMarco et al., 2024). For families of regular polynomial endomorphisms on $\Phi(s,z)=(s,f_s(z))$48, stronger degree-stabilization statements require additional hypotheses controlling the line at infinity and excluding certain critical-orbit coincidences (Zhong, 19 Aug 2025). In the affine plane, the full equivalence between infinitely many periodic points on a curve and time-reversal symmetry remains conjectural (Dujardin et al., 2014).

The accumulated evidence nevertheless points to a coherent principle. Across families on projective space, toric compactifications, affine polynomial automorphisms, and orbit-relative problems on $\Phi(s,z)=(s,f_s(z))$49, dense sets of dynamically special points do not occur arbitrarily. They are forced by hidden geometric structure: maximal Green-current rank, quotient-torus torsion geometry, reversibility, diagonal-preimage relations, or Frobenius-type periodicity (DeMarco et al., 2024, Lin, 2017, Dujardin et al., 2014, Schmidt, 2020, Xie, 2016). The Relative Dynamical Manin–Mumford Conjecture is the attempt to express this principle in a single invariant language for algebraic families of dynamical systems.