Residual periodicity on the Markoff surface
Number Theory
2016-05-05 v2 Algebraic Geometry
Dynamical Systems
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.
Cite
@article{arxiv.1504.07099,
title = {Residual periodicity on the Markoff surface},
author = {Solomon Vishkautsan},
journal= {arXiv preprint arXiv:1504.07099},
year = {2016}
}
Comments
13 pages, 1 table Accepted on Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl