A note on reversibility and Pell equations

TL;DR

This paper explores the connection between reversibility in hyperbolic toral automorphisms and solutions to generalized Pell equations, analyzing how common reversibility is and characterizing the generic case.

Contribution

It establishes a link between reversibility of hyperbolic toral automorphisms and Pell equations, providing new insights into their structure and prevalence.

Findings

Reversibility is linked to solutions of a generalized Pell equation.

Reversibility is not a universal feature among these automorphisms.

The paper characterizes the generic setting for reversibility.

Abstract

We consider hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. We will prove that within this context reversibility is linked to a generalized Pell equation whose solutions we will analyze. Additionally, we will verify to what extent reversibility is a common feature and characterize the generic setting.