Reversibility and symmetry of affine toral automorphisms

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.