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Reversibility and symmetry of affine toral automorphisms

Kuntal Banerjee, Anubrato Bhattacharyya, Krishnendu Gongopadhyay, Subhamoy Mondal

TL;DR

This work addresses reversibility and strong reversibility for affine toral automorphisms $f_{A,\bar a}(\bar x)=A\bar x+\bar a$ on $\mathbb{T}^2$, revealing that the presence of the eigenvalue $1$ in $A$ crucially affects the obstructions to reversibility. It provides explicit criteria: reversibility requires a reversing symmetry $R$ with $RAR^{-1}=A^{-1}$ and a linear congruence $(AR+I)\bar a+(A-I)\bar r\equiv0$, with the latter always solvable when $1$ is not an eigenvalue, while it imposes a cohomological obstruction when $1$ is an eigenvalue. A sharp fixed-point criterion via Pick's theorem yields a gcd-based, area-inequality condition ensuring fixed points for all translations, and the entropy of affine maps matches that of their linear parts. The paper also shows a dichotomy in the number of conjugacy classes within a similarity class: finite when $\det(A-I)\neq0$ and uncountable when $1$ is an eigenvalue. Finally, it interprets reversibility cohomologically and provides a complete classification of reversible affine toral automorphisms, with explicit translation conditions and a cohomological perspective guiding extensions to higher dimensions.

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.

Reversibility and symmetry of affine toral automorphisms

TL;DR

This work addresses reversibility and strong reversibility for affine toral automorphisms on , revealing that the presence of the eigenvalue in crucially affects the obstructions to reversibility. It provides explicit criteria: reversibility requires a reversing symmetry with and a linear congruence , with the latter always solvable when is not an eigenvalue, while it imposes a cohomological obstruction when is an eigenvalue. A sharp fixed-point criterion via Pick's theorem yields a gcd-based, area-inequality condition ensuring fixed points for all translations, and the entropy of affine maps matches that of their linear parts. The paper also shows a dichotomy in the number of conjugacy classes within a similarity class: finite when and uncountable when is an eigenvalue. Finally, it interprets reversibility cohomologically and provides a complete classification of reversible affine toral automorphisms, with explicit translation conditions and a cohomological perspective guiding extensions to higher dimensions.

Abstract

We study reversibility and strong reversibility of affine automorphisms of the two-torus, written as . We derive explicit criteria for the reversibility of such maps in terms of the matrix and the translation . If is not an eigenvalue of , reversibility of the affine map coincides with reversibility of . When 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.
Paper Structure (14 sections, 16 theorems, 78 equations, 3 figures)

This paper contains 14 sections, 16 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine reversibility
  4. Reversibility in $GL(2,\mathbb Z)$ and $PGL(2,\mathbb Z)$
  5. Reversibility in $PGL(2,\mathbb Z)$
  6. Strong reversibility
  7. Fixed Points and Basic Dynamical Properties
  8. Fixed-point criterion
  9. Number of Conjugacy classes in a Similarity Class
  10. Case 1: $1$ is not an eigenvalue of $A$.
  11. Case 2: $1$ is an eigenvalue of $A$.
  12. Reversible and Strongly Reversible Elements of $GL(2,\mathbb{Z})\ltimes \mathbb{T}^{2}$
  13. Classification of Reversible Affine Toral Automorphisms
  14. Cohomological interpretation of the reversibility condition

Key Result

Theorem 2.2

Let the vertices of a polygon $S$ have all its co-ordinates as integers. Then the area of $S$ is given by where $k_1, k_2$ denote respectively the number of interior and boundary (including both the vertices and points along the sides) points with integer co-ordinates.

Figures (3)

  • Figure 1: Illustration of Pick's Theorem
  • Figure 2: Toral endomorphism for the Arnold's cat map induced by $A=2111$. Left figure depicts the image $A([0,1]^2)$ in $\mathbb{R}^2$, while the right figure depicts the same image after taking $\text{mod}~ \mathbb{Z}^2$.
  • Figure 3: Recovering a representative of $-\overline{b}$ inside the parallelogram.

Theorems & Definitions (39)

  • Definition 2.1
  • Theorem 2.2: Pick's Theorem
  • Remark 1
  • Definition 2.3
  • Lemma 3.1: IS
  • Theorem 3.2: IS
  • Remark 2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 29 more