Mathematical Background

This page summarizes the definitions behind the data in Dynabase and the open problems the examples are meant to test. Silverman's The Arithmetic of Dynamical Systems (Graduate Texts in Mathematics 241, Springer, 2007) is a standard introduction to the subject.

Iteration and orbits

A dynamical system is a set S together with a map f from S to itself, which can be applied over and over. Write fⁿ for the n-th iterate: f composed with itself n times, with f⁰ the identity. The (forward) orbit of a point P is the sequence P, f(P), f²(P), …. The point is

periodic if the orbit returns to P: fⁿ(P) = P for some n ≥ 1, the smallest such n being its (minimal) period;
preperiodic if its orbit is finite, i.e. some iterate of P is periodic;
wandering if its orbit is infinite.

A preperiodic point that is not periodic sits on a tail leading into a cycle. A preperiodic point has type (m, n) when it takes m steps to reach a cycle of length n.

Arithmetic dynamics studies the case where S is algebraic, typically the points of a variety over a number field K, and f is given by polynomials with coefficients in K. The questions are about the points defined over K itself.

Example. Take f(z) = z² − 1 over ℚ. Then 0 → −1 → 0 is a cycle of length 2, the point 1 maps into it (1 → 0), and the point at infinity is fixed. These four points are all of the rational preperiodic points of f; every other rational number, such as 2 → 3 → 8 → …, wanders off to infinity. Drawing an arrow from each point to its image turns the preperiodic points into a directed graph, and this graph is what Dynabase stores.

Maps of projective space

Dynabase works with self-maps of projective space. This is where the computational tools are available (the algorithms, and their implementations in SageMath, are almost all written for projective space), and many of the main conjectures of the subject reduce to this case. A map from P^N to itself of degree d is given by N + 1 homogeneous polynomials of degree d with no common factor. It is a morphism (defined at every point) when the polynomials also have no common zero. On the projective line, N = 1, every such map is a morphism. The site displays a map of the projective line as (F₀(x, y) : F₁(x, y)), which is the rational function F₀(z, 1) / F₁(z, 1) on the affine coordinate z = x/y. A polynomial f(z) of degree d appears as (y^df(x/y) : y^d).

Conjugation, moduli and models

Changing coordinates by an invertible linear map α of projective space (an element of PGL_N+1) replaces f by its conjugate f^α = α⁻¹ ∘ f ∘ α. Iterating commutes with conjugating, (f^α)ⁿ = (fⁿ)^α, so α carries orbits of one map to orbits of the other: the two maps have the same dynamics, written in different coordinates. The conjugacy classes of degree d maps form the moduli space M_d.

Dynabase has one entry per conjugacy class. The different polynomial representatives of a class are called models, and a single entry can store several of them:

Original: the model as it was given by its source.
Monic centered: a map is conjugate to a polynomial exactly when it has a totally ramified fixed point. Moving that point to infinity and normalizing so that the leading coefficient is 1 and the next coefficient is 0 gives the usual normal form of a polynomial. This may need a field extension.
Reduced: the model with the smallest resultant and, among those, the smallest coefficients, found by the algorithm of Hutz and Stoll. It is determined by the conjugacy class.
Chebyshev and Newton: a map is a Chebyshev map if it is conjugate to a Chebyshev polynomial, and a Newton map if it is Newton's method z − F(z)/F′(z) for some polynomial F.

Conjugation does not always respect the field of definition. Two maps defined over K can be conjugate only over a larger field; such maps are twists of each other. Their rational points can behave differently, so twists are separate entries in Dynabase, linked to each other. The maps α that conjugate f to itself form its automorphism group, which is finite for a morphism. Faber, Manes and Viray gave an efficient algorithm to compute it on the projective line.

Multiplier invariants and labels

If Q is a fixed point of f, its multiplier is the derivative of f at Q, computed in a local coordinate. Multipliers do not change under conjugation, and neither do the elementary symmetric functions of the multipliers of all the fixed points, σ₁⁽¹⁾, …, σ_d+1⁽¹⁾. The fixed points of the iterate fⁿ give invariants σ⁽ⁿ⁾ in the same way. The multipliers themselves may only be defined over an extension, but these symmetric functions always lie in the field of definition of f. By a theorem of McMullen, the invariants of enough periods pin down a conjugacy class up to finitely many possibilities, apart from the Lattès maps.

This gives each conjugacy class its label dim.sig1.sig2.ord: the dimension, then 8-character SHAKE-256 hashes of σ⁽¹⁾ and σ⁽²⁾, then an ordinal that tells apart the classes sharing those invariants. Because the label is built from invariants, any model of the map leads to the same label.

Fields and bad reduction

The field of definition of a model is the field generated by its coefficients. Number fields are named by their LMFDB labels; a field not in the LMFDB is shown by its defining polynomial. The resultant of a model is the resultant of its defining polynomials. It is nonzero exactly when the model is a morphism, and the primes dividing it are the primes where the model has bad reduction.

The problems

Rational preperiodic points

By Northcott's theorem, a morphism of degree at least 2 has only finitely many preperiodic points over a given number field, because every preperiodic point has bounded height. Morton and Silverman conjectured that much more is true: the number is bounded uniformly, independent of the particular map and field.

Uniform Boundedness Conjecture (Morton–Silverman, 1994). For all integers N ≥ 1, d ≥ 2 and D ≥ 1 there is a constant C(N, d, D) such that for every number field K with [K : ℚ] ≤ D and every morphism f : P^N → P^N of degree d defined over K, the number of preperiodic points of f in P^N(K) is at most C(N, d, D).

The conjecture is strong. Applied to the degree 4 Lattès maps coming from multiplication by 2 on elliptic curves, it already implies the uniform boundedness of torsion on elliptic curves (Mazur's theorem over ℚ, Merel's over number fields).

The conjecture is open even for quadratic polynomials over ℚ. Poonen showed that if no z² + c has a rational cycle of length more than 3, then such a map has at most 9 rational preperiodic points, counting the point at infinity, and he listed the possible graphs. Cycles of length 4 and 5 have been ruled out (Morton; Flynn, Poonen and Schaefer), and length 6 is ruled out assuming standard conjectures on elliptic curves and abelian varieties (Stoll). In the other direction, Doyle and Hyde showed that polynomials of degree d over ℚ can have at least d + max(6, ⌊log₂ d⌋) rational preperiodic points, so any uniform bound must grow faster than d itself.

Two special cases get their own attention: long rational cycles and long rational tails. Interpolation easily produces cycles and tails whose length grows linearly with the degree, so the question is how far beyond that one can go. Bounds on both lengths, together with the known (but not explicit) uniform bound on rational preimages, would lead to an explicit Morton–Silverman constant. In the genetic algorithm data of Hutz (2026), no rational cycle was longer than d + 3 for polynomials or 2d + 3 for rational maps, and no rational tail was longer than d + 2 or 2d + 2 respectively.

Postcritically finite maps

The critical points of a map are the points where it fails to be locally one-to-one. A map is postcritically finite (PCF) if every critical point is preperiodic. The critical portrait is the directed graph formed by the critical points and everything in their forward orbits. By Thurston's rigidity theorem, PCF maps other than the flexible Lattès maps (which come in one-parameter families) are isolated points of moduli space, so they are defined over number fields. Their heights are also bounded, which gives finiteness:

Theorem (Ingram, 2012, for polynomials; Benedetto, Ingram, Jones and Levy, 2014, for rational maps). For every d ≥ 2 and B ≥ 1, there are only finitely many conjugacy classes of PCF maps of degree d, other than flexible Lattès maps, defined over number fields of degree at most B.

Both proofs show that PCF maps form a set of bounded height in moduli space, and Northcott's theorem then leaves only finitely many over fields of bounded degree. Ingram's bound is effective, so for polynomials the finite list can in principle be computed. For small degrees the complete lists over ℚ are known: quadratic polynomials (only z², z² − 1 and z² − 2, up to conjugacy; Ingram), cubic polynomials (Anderson, Manes and Tobin), quartic polynomials (Fraser) and quadratic rational maps (Lukas, Manes and Yap).

Small canonical heights

The height of a rational point measures its arithmetic size; for a rational number a/b in lowest terms it is log max(|a|, |b|). Applying a map of degree d multiplies heights by roughly d, and the canonical height ĥ_f(P) = lim h(fⁿ(P)) / dⁿ makes this exact: ĥ_f(f(P)) = d ĥ_f(P), and ĥ_f differs from the ordinary height by a bounded amount. Over a number field, ĥ_f(P) = 0 exactly when P is preperiodic, and Northcott's theorem guarantees a wandering point with the smallest positive canonical height.

How small can that be? Conjugating a map can make canonical heights as small as we like, so the height of a point has to be compared with the size of the map itself, measured by a height h_M(f) on the moduli space of degree d maps. Lang conjectured that on an elliptic curve the canonical height of a non-torsion point is bounded below in terms of the curve's minimal discriminant; the dynamical version replaces the discriminant with this height on moduli space.

Dynamical Lang Conjecture (Silverman, The Arithmetic of Dynamical Systems, Conjecture 4.98). Let K be a number field and d ≥ 2. There is a constant c = c(K, d) > 0 such that for every map f : P¹ → P¹ of degree d defined over K and every wandering point P ∈ P¹(K), ĥ_f(P) ≥ c · h_M(f).

Looper proved bounds of this shape for polynomials, under good reduction hypotheses and, for all polynomials, assuming the abcd-conjecture; the constants are not explicit.

Dynabase records the height ratio ĥ_f(P) / h_M(f), with h_M(f) the largest height of the fixed point invariants σ_i⁽¹⁾, for the point with the smallest ratio known. For quadratic maps these invariants give coordinates on moduli space (Milnor), and for polynomials they determine the class up to finitely many choices. For rational maps of higher degree they do not, so for those maps this measure is only a stand-in for a moduli height, but using the same measure in every degree makes the examples comparable. Earlier searches covered quadratic polynomials, cubic polynomials and quadratic rational maps (Benedetto et al.). The genetic algorithm data of Hutz (2026) suggests that the smallest ratio over ℚ shrinks roughly like 10^−3d/2 for polynomials and 10^−2d for rational maps of degree d.

Automorphism groups

The automorphism group Aut(f) of a map f : P^N → P^N consists of the changes of coordinates α ∈ PGL_N+1 that conjugate f to itself, f^α = f.

Theorem (Petsche, Szpiro and Tepper, 2009). If f : P^N → P^N is a morphism of degree d ≥ 2, then Aut(f) is a finite subgroup of PGL_N+1.

So the possible automorphism groups are among the finite subgroups of PGL_N+1, which over an algebraically closed field of characteristic 0 have been classified in the first few dimensions:

PGL₂ (the projective line): the cyclic and dihedral groups, and the symmetry groups A₄, S₄ and A₅ of the Platonic solids. This goes back to Klein (1884); Silverman (1995) gives it in modern form, with explicit generators.
PGL₃ (the projective plane): Blichfeldt (1917), also in Miller, Blichfeldt and Dickson (1916). Besides the groups coming from lower dimensions and from monomial matrices, there are finitely many primitive groups, among them A₅, A₆, PSL₂(F₇) of order 168 and the Hessian group of order 216.
PGL₄ (projective 3-space): Blichfeldt (1905), and again in his 1917 book.

De Faria and Hutz showed how to build maps with a prescribed automorphism group and that every finite subgroup is the automorphism group of infinitely many maps. What is less clear is how simple such a map can be. The problem is to find, for each possible group, maps of the smallest possible degree, defined over the smallest possible field, whose automorphism group is exactly that group and not a larger one. The field matters because a map defined over ℚ can have automorphisms that are only defined over an extension. De Faria and Hutz gave a map over ℚ for every group except the tetrahedral group A₄, and stated that none exists over ℚ; Gontmacher, Hutz, Jorgenson, Srimani and Xu (2021) then found one, (z³ − 3) / (−3z²), so every finite subgroup of PGL₂ is the automorphism group of some map defined over ℚ. Dynabase records the automorphism group of its maps, computed over the algebraic closure, when it can be computed.

For polynomials the answer is complete (see Fujimura and Nishizawa on the symmetry loci of polynomial maps). Let f be a polynomial of degree d ≥ 2.

If f is conjugate to z^d, then Aut(f) is the dihedral group of order 2(d − 1): the rotations z ↦ ζz with ζ^d−1 = 1, and z ↦ 1/z, which swaps the two totally ramified fixed points 0 and ∞.
Otherwise ∞ is its only totally ramified fixed point, so every automorphism fixes ∞ and is affine. Writing f in monic centered form, an automorphism must also fix 0, so it is z ↦ ζz, and this commutes with f exactly when ζ^e−1 = 1 for every exponent e appearing in f. So Aut(f) is cyclic of an order m dividing d − 1, and f has an automorphism of order m exactly when its monic centered form is z·g(z^m).

So in degree d the possible automorphism groups of a polynomial are the trivial group, the cyclic group C_m for each divisor m > 1 of d − 1 (for example z^d + z^m+1, or z^d + z when m = d − 1), and the dihedral group of order 2(d − 1), for z^d alone. All of these are defined over ℚ, and the tetrahedral, octahedral and icosahedral groups never occur. Dynabase has an example of every one of these groups in each degree up to 10.

Status of the problems

The charts on the Status of Problems page rate each problem, degree and field:

proven: the complete answer is a theorem, so the data is a complete list;
conjectural: a complete answer has been conjectured, backed by partial proofs and extensive searches;
experimental: the examples come from computer searches, with no conjectured complete answer;
uncomputed: the complete answer is a theorem, but the database does not have its examples for that degree yet;
open: little or no systematic data is available yet.

The automorphism groups chart also gives, in each cell, the number of distinct nontrivial automorphism groups among the maps of that degree, type and field in the database.

Dynabase