A Genetic Algorithm for Generating Extreme Examples in Arithmetic Dynamics

TL;DR

This paper introduces a genetic algorithm to generate extreme examples in arithmetic dynamics, expanding known behaviors and laying groundwork for future machine learning applications.

Contribution

It presents a novel genetic algorithm approach to find extreme arithmetic dynamical systems, covering polynomials up to degree 13 and rational functions up to degree 5.

Findings

Generated numerous new extreme examples in arithmetic dynamics.

Expanded the known set of behaviors for conjectured dynamical properties.

Provided data for polynomials and rational functions of specified degrees.

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.