A Genetic Algorithm for Generating Extreme Examples in Arithmetic Dynamics
Benjamin Hutz
TL;DR
This work demonstrates a straightforward genetic-algorithm pipeline to produce extreme examples in arithmetic dynamics, applying it to four concrete problems that probe canonical heights, preperiodic structures, and tail/cycle lengths across polynomials up to degree 13 and rational maps up to degree 5. By encoding maps via their orbits and scoring generations with problem-specific fitness functions, the approach yields extreme instances that match or exceed previously known benchmarks in several cases and reveals patterns such as dynamical compression in many extreme examples. The data set, architecture, and code provided lay a foundation for integrating more advanced ML/RL techniques to push beyond current degree limits and to facilitate rapid discovery of extreme dynamical behavior. The results expand the catalog of extreme phenomena in arithmetic dynamics and offer a practical path toward quantitative insights into conjectures like Morton–Silverman Uniform Boundedness and dynamical Lang-type height bounds.
Abstract
We describe a genetic algorithm to find extreme examples in the arithmetic of dynamical systems. The algorithm is applied to four problems: small (non-zero) canonical heights, many rational preperiodic points, long rational cycles, and long rational tails. Data is provided for extreme examples generated for polynomials up to degree 13 and rational functions up to degree 5. This work significantly expands the known examples of extreme behavior for several of the conjectured behaviors in arithmetic dynamics and provides a foundation from which to begin a more advanced application of machine learning techniques in the creation of extreme examples for arithmetic dynamics.