On the dynamics of the Pappus-Steiner map

TL;DR

This paper introduces a new two-dimensional dynamical system derived from classical projective geometry theorems, analyzes its properties, and characterizes prime fields where periodic points exist, leading to a Galois-theoretic conjecture.

Contribution

It formulates an explicit dynamical system from Pappus and Steiner theorems, studies its properties, and connects prime periodic points to Galois theory, also relating Leisenring's theorem to the same system.

Findings

Explicit formula for the dynamical system derived from classical theorems.

Characterization of primes with periodic points using Artin reciprocity.

Connection between geometric theorems and Galois-theoretic conjectures.

Abstract

We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin reciprocity to characterise all sufficiently large primes $p$ for which this system admits periodic points of orders $3$ and $4$ over the field ${\mathbb F}_p$; this leads to an unexpected Galois-theoretic conjecture for $n$-periodic points. We also give a short discussion of Leisenring's theorem, and show that it leads to the same dynamical system as the Pappus-Steiner theorem. The appendix contains a computer-aided analysis of this system over the field of real numbers.