On the dynamics of the Pappus-Steiner map

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.