About Dynabase
Dynabase (the Database of Arithmetic Dynamics) is a collection of extreme examples in arithmetic dynamics: maps whose rational points behave as unusually as anyone has managed to find. Many of the central conjectures in the subject predict that some quantity is bounded — how many rational preperiodic points a map can have, how long a rational cycle can be, how small the canonical height of a wandering point can be. The best way to test such a bound, and to guess what its true value should be, is to know the most extreme examples. Those examples are currently scattered across papers, theses, personal web pages and computer searches. Dynabase gathers them in one place, stores them in a uniform format, and computes the same set of invariants for every one of them.
The problems
Dynabase collects extreme examples for four problems:
- Rational preperiodic points. How extreme can the rational preperiodic points of a map
be, given its degree and field of definition?
- Long cycles: maps with a rational periodic cycle as long as possible.
- Long tails: maps with a rational point that takes as many steps as possible to reach a cycle.
- Many points: maps with as many rational preperiodic points in total as possible.
- Postcritically finite maps. Maps whose critical points all have finite orbits, which are rare: find all of them, together with their critical portraits.
- Small canonical heights. Maps with a wandering rational point whose canonical height is as small as possible compared to the size of the map.
- Automorphism groups. For each group that can occur as the automorphism group of a map, find maps of the smallest possible degree, defined over the smallest possible field, whose automorphism group is exactly that group (see de Faria and Hutz).
The Mathematical Background page explains each problem and what is known about it.
What is in the database
The current release covers maps of the projective line (dimension 1): polynomials and rational functions of degree 2 through 15, defined over the rational numbers or over quadratic number fields. Each entry is a conjugacy class of maps, so two maps that differ only by a change of coordinates are stored once. For every entry, Dynabase records:
- several models of the map: the one given in its source, and when they can be computed a monic centered model (for polynomials) and a reduced model;
- its field of definition (with a link to the LMFDB when the field is there), resultant and primes of bad reduction;
- a label built from its multiplier invariants, which identifies the conjugacy class;
- its rational preperiodic points, as a directed graph, over its own field and over any other field it was studied over;
- whether it is postcritically finite, and if so its critical portrait;
- its automorphism group, rational twists, and whether it is a Chebyshev, Newton or Lattès map;
- the smallest canonical height ratio known for one of its rational points;
- the sources it comes from.
The examples come from the published literature, from computer searches and from personal communications; each one is credited to its source. The Summary of Included Data lists every source and counts the maps, and Status of Problems shows how much is known about each problem in each degree. Every invariant is computed in SageMath. The Mathematical Background page explains the terms used on this site.
Using the site
The selectors at the top of every page choose a dimension, a degree, polynomial or rational maps, and a problem:
- Rational Preperiodic: maps with many rational preperiodic points, long rational cycles or long rational tails. Each table lists one map for each distinct graph of rational preperiodic points, with the most points first.
- Postcritically Finite: maps whose critical points all have finite orbits, with their critical portraits. Where a complete classification is known, the table is the complete list.
- Small Height Ratio: maps with a wandering rational point of very small canonical height, compared to the size of the map. Each table is sorted from the smallest ratio.
- Automorphism Groups: maps with a nontrivial automorphism group, largest group first.
Tables are split by the degree of the field the data is over. The Summary of Extreme Examples lists the most extreme map in the database for each problem, in each degree.
Dynabase is maintained by Benjamin Hutz (Saint Louis University). Corrections and new examples are welcome.
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