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Abstract

A hyperbolic transformation of the torus is an example of a function that is Devaney chaotic; that is, it is topologically transitive and has dense periodic points. An irrational rotation of the torus, on the other hand, is not chaotic because it has no periodic points. We show that a hyperbolic transformation of the torus followed by a translation (an affine hyperbolic toral automorphism) has dense periodic points and maintains transitivity. As a consequence, affine toral automorphisms are chaotic, even when the translation is an irrational rotation.