A class of differentiable toral maps which are topologically mixing

Abstract

We show that on the 2-torus T 2 \mathbb {T}^{2} there exists a C 1 C^{1} open set U \mathcal {U} of C 1 C^{1} regular maps such that every map belonging to U \mathcal {U} is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of C 1 C^{1} toral diffeomorphisms, but that the property does hold for the class of C 1 C^{1} diffeomorphisms on the 3-torus T 3 \mathbb {T}^{3} . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of C 1 C^{1} diffeomorphisms on the n n -torus T n \mathbb {T}^{n} ( n ≥ 4 n\ge 4 ).