A class of differentiable toral maps which are topologically mixing

Sumi, Naoya

doi:10.1090/S0002-9939-99-04608-0

A class of differentiable toral maps which are topologically mixing
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Proc. Amer. Math. Soc. 127 (1999), 915-924 Request permission

Abstract:

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

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