Reversibility and symmetry of affine toral automorphisms

TL;DR

This paper investigates the conditions under which affine automorphisms of the two-torus are reversible or strongly reversible, providing explicit criteria, geometric conditions, and analyzing their dynamical properties including entropy and conjugacy classes.

Contribution

It offers explicit criteria for reversibility of affine toral automorphisms, including cases with eigenvalue 1, and characterizes their dynamics, fixed points, and conjugacy classes.

Findings

Reversibility depends on matrix $A$ and translation $ar{a}$.

Additional arithmetic obstructions occur when 1 is an eigenvalue.

A geometric condition based on Pick's Theorem guarantees fixed points.

Abstract

We study reversibility and strong reversibility of affine automorphisms of the two-torus, written as $f_{A,\bar{a}}(\bar{x})=A\bar{x}+\bar{a} \ (\mathrm{mod}\ \mathbb{Z}^2)$. We derive explicit criteria for the reversibility of such maps in terms of the matrix $A$ and the translation $\bar{a}$. If $1$ is not an eigenvalue of $A$, reversibility of the affine map coincides with reversibility of $A$. When $1$ is an eigenvalue, additional arithmetic obstructions appear. We also provide a simple geometric condition, based on Pick's Theorem, that guarantees the existence of fixed points, along with a description of the dynamics of affine toral automorphisms. We also compute the entropy and characterize when conjugacy classes in the affine group are finite or uncountable.