Reversibility and symmetry of affine toral automorphisms
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915747216752640 |
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| author | Banerjee, Kuntal Bhattacharyya, Anubrato Gongopadhyay, Krishnendu Mondal, Subhamoy |
| author_facet | Banerjee, Kuntal Bhattacharyya, Anubrato Gongopadhyay, Krishnendu Mondal, Subhamoy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15827 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reversibility and symmetry of affine toral automorphisms Banerjee, Kuntal Bhattacharyya, Anubrato Gongopadhyay, Krishnendu Mondal, Subhamoy Dynamical Systems 11F06, 15B36, 20E45, 37B05 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. |
| title | Reversibility and symmetry of affine toral automorphisms |
| topic | Dynamical Systems 11F06, 15B36, 20E45, 37B05 |
| url | https://arxiv.org/abs/2601.15827 |