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arXiv · 2601.15827

Reversibility and symmetry of affine toral automorphisms

Kuntal Banerjee·Anubrato Bhattacharyya·Krishnendu Gongopadhyay·Subhamoy Mondal
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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.

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Kuntal Banerjee, Anubrato Bhattacharyya, Krishnendu Gongopadhyay, Subhamoy Mondal. 2026-01-22. Reversibility and symmetry of affine toral automorphisms. https://arxiv.org/abs/2601.15827

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