Exact realization of prescribed dynamics on prescribed postcritical sets of rational maps (DeMarco-Koch-McMullen, Question 1.2)

This problem is the open core of Question 1.2 of DeMarco, Koch, and McMullen, 'On the postcritical set of a rational map' (Math. Ann. 377 (2020), 1-18). The source proved that every finite set of algebraic numbers on the Riemann sphere occurs as the postcritical set of a rigid postcritically finite rational map, and asked the sharper question of which self-maps of a prescribed finite algebraic set are induced by such maps. The question is settled for sets of two and three points in the source itself; it remains open for every set of four or more prescribed algebraic points, including the source's explicit test instance $X=\{0,1,4,\infty\}$ with $F=\mathrm{id}$. Later work resolves only the location-free combinatorial analog for polynomials (Floyd-Kim-Koch-Parry-Saenz, 2026), gives a conditional reduction for four-point sets via the composition structure of self-maps (Ma, 2023), and develops perturbative and transcendental analogs (Bishop-Lazebnik; Bishop-Lazebnik-Urbanski; Mukundan-Prochorov-Reinke). The problem asks for a classification of exactly which pairs $(X,F)$ are realized by a rigid rational map $f$ with $P(f)=X$ and $f|_{P(f)}=F$; the companion degree-effectiveness questions raised by the same source (the degree functions $B(X)$ and $D(X)$, and minimal passport extensions) also remain unaddressed in the later literature.

Background

A rational map $f:\mathbb{P}^1\to\mathbb{P}^1$ of degree $d\ge2$ is a quotient of polynomials acting on the Riemann sphere. Its critical points $C(f)$ are the points at which $f$ is not locally injective (the zeros of its derivative), and their images are the critical values $V(f)$. The postcritical set is the forward orbit of the critical values,

$P(f)=\bigcup_{n\ge0}f^n(V(f)),$

that is, the smallest forward-invariant subset of $\mathbb{P}^1$ containing $V(f)$. The map $f$ is postcritically finite (PCF) if $P(f)$ is finite; PCF maps are the combinatorially best-understood objects of complex dynamics and occupy a central place in arithmetic dynamics.

A PCF map is called rigid if every PCF map uniformly close to it, with the same number of postcritical points, is conformally conjugate to it. By Thurston's topological characterization of rational maps (proved by Douady and Hubbard), a postcritically finite branched covering of the sphere is equivalent to a rational map precisely when it carries no Thurston obstruction (a finite system of curves with a certain contraction property), and the only PCF rational maps that fail rigidity are the flexible Lattes maps — maps descended from multiplication by an integer on a complex torus, followed by a continuously deformable translation; all of them have $|P(f)|=4$ and Julia set (the closed set where the iterates of $f$ fail to form a normal family) equal to the whole sphere. Rigid PCF maps are conformally conjugate to maps defined over $\overline{\mathbb{Q}}$, so their postcritical sets are finite sets of algebraic numbers.

DeMarco, Koch and McMullen proved the converse to this last statement (their Theorem 1.1): every finite set $X\subset\mathbb{P}^1(\overline{\mathbb{Q}})$ with $|X|\ge2$ is the postcritical set of a rigid PCF rational map. The proof composes a polynomial $g$ with prescribed critical values $V_0(g)=X$ with a Belyi polynomial — a polynomial all of whose critical values lie in $\{0,1\}$, whose existence for any finite set of algebraic numbers is the content of Belyi's theorem from arithmetic geometry. This realizes the set $X$, but leaves the induced dynamics uncontrolled (the construction gives $f(X)\subset X$, while which self-map of $X$ results is dictated by the Belyi data), and the degrees produced are potentially enormous: for the degree functions $B(X)$ (the minimum degree of a Belyi polynomial $\beta$ with $\beta(X)\subset\{0,1\}$) and $D(X)$ (the minimum degree of a polynomial $f$ whose finite postcritical set is $X$), the paper proves only $D(X)\le B(X)+|X|+1$ and observes that lower bounds 'seem hard to come by.'

The sharper question — Question 1.2 of the same paper — asks to realize the dynamics as well as the set: given a finite set $X\subset\mathbb{P}^1(\overline{\mathbb{Q}})$, is every map $F:X\to X$ induced as $F=f|_{P(f)}$ by a rigid rational map $f$ with $P(f)=X$? For $|X|=2$ the answer is no (maps with two postcritical points are conjugate to $z\mapsto z^{\pm d}$, so only bijections arise), and for $|X|=3$ the answer is yes (all seven self-maps, up to permuting the three points, are realized explicitly). If small perturbations of the set are allowed, everything is realizable: their Theorem 1.3 (strengthened in Theorem 6.1 to also prescribe the multiplicities, that is, the local degrees of $f$ at the postcritical points) produces sequences of rigid PCF maps $f_n$ with $|P(f_n)|=|X|$, $P(f_n)\to X$ and $f_n|_{P(f_n)}\to F$, using contraction of a pullback map on the Teichmuller space of the punctured sphere (the universal cover of the moduli space of $n$-point spheres). A companion result of Hurwitz type shows that any finite list of partitions — the 'passport' datum recording, for each critical value of a degree-$d$ map, the multiset of local degrees in its fiber — can be extended to the passport of a rational map.

The obstacle to exactness is arithmetic. For four-point sets, rigid Lattes maps realize postcritical sets $\{0,1,\infty,z\}$ only for $z$ in a dense but very small subset $L\subset\overline{\mathbb{Q}}$ (for instance $L\cap\mathbb{Z}=\{-1,2\}$), and a second construction — iterating an explicit map on the moduli space $\mathcal{M}_{0,4}\cong\mathbb{P}^1\setminus\{0,1,\infty\}$ whose periodic points correspond to PCF maps — realizes another dense set $K$; these two loci intersect only finitely, and neither hits an arbitrary prescribed algebraic cross-ratio. The paper states that for $|X|=4$ most cases are 'simply open,' highlights $X=\{0,1,4,\infty\}$ with $F=\mathrm{id}$ as an instance whose answer is unknown, and nothing general is known for larger sets. Later work has resolved the location-free combinatorial analog of this question for polynomials and has produced conditional reductions for four-point sets, but the prescribed-location question itself — the subject of this problem — remains open.

Problem Statement

Determine, for every finite set $X\subset\mathbb{P}^1(\overline{\mathbb{Q}})$ with $|X|\ge4$ and every map $F:X\to X$, whether there exists a rigid rational map $f$ of degree at least 2 such that

$P(f)=X\quad\text{and}\quad f|_{P(f)}=F.$

Equivalently: classify the realizable pairs $(X,F)$ among finite algebraic sets of at least four points. A complete answer either proves that every such pair is realized or exhibits the unrealizable pairs and proves their non-realizability. This includes the source's explicit test instance $X=\{0,1,4,\infty\}$ with $F=\mathrm{id}_X$. The cases $|X|\le3$ were settled in the source itself (negative for $|X|=2$ unless $F$ is bijective; positive for $|X|=3$) and are not part of the open core. Two companion effectiveness issues raised by the same source remain open and are recorded here as context only, outside the completion boundary of this problem: the sharp behavior of the degree functions $B(X)$ and $D(X)$ (for which no lower bounds are known), and the characterization of minimal passport extensions in the Hurwitz-type Theorem 5.1.

References

1. Laura G. DeMarco, Sarah C. Koch, Curtis T. McMullen, On the postcritical set of a rational map, Mathematische Annalen 377 (2020), no. 1-2, 1-18 (published online 2018). DOI: 10.1007/s00208-018-1732-6, https://doi.org/10.1007/s00208-018-1732-6; preprint arXiv:1709.06866, https://arxiv.org/abs/1709.06866.
2. William Floyd, Daniel Kim, Sarah Koch, Walter Parry, Edgar Saenz, Realizing polynomial ramification portraits, Ergodic Theory and Dynamical Systems 46 (2026), no. 7, 1710-1734. DOI: 10.1017/etds.2026.10279, https://doi.org/10.1017/etds.2026.10279; preprint (under the title 'Realizing polynomial portraits') arXiv:2105.10055, https://arxiv.org/abs/2105.10055.
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