Small perturbations of the automorphisms of the torus

Abstract

Let the torus T ² be realized in the form of the unit square in the plane (x ₁, x ₂) with pairwise identified sides. A transformation $x \rightarrow Ax = \bar{x}$, given by the integer valued matrix $A = \| a_{ij} \|$ with determinant ±1 acting on the torus according to the formula $\bar{x}_{i} = \sum_{j} a_{ij} x_{j} \text{ (mod 1)}$, i = 1, 2, is called an automorphism of T ².