Residual periodicity on the Markoff surface

TL;DR

This paper investigates the arithmetic dynamics of the Markoff surface over rationals, revealing strong residual periodicity linked to specific conic sections lacking rational points, and shows how removing these sections affects the dynamics.

Contribution

It demonstrates that the composition of reflections on the Markoff surface exhibits strong residual periodicity due to certain conic sections, and explains how removing these sections alters the dynamical behavior.

Findings

The dynamical system is strongly residually periodic.

Periodic conic sections explain residual periodicity.

Removing these conic sections eliminates the residual periodicity.

Abstract

A case study of arithmetic dynamics over the rationals on the Markoff surface is presented, in particular the local-global dynamical property of strong residual periodicity. The dynamical system induced by the composition of any two of the reflections from the three special points at infinity on the Markoff surface is shown to be strongly residually periodic. This residual periodicity is explained by the existence of periodic conic sections of the Markoff surface with no rational points. It is also proven that cutting these conic sections from the surface eliminates strong residual periodicity.