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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A class of differentiable toral maps
which are topologically mixing

Author: Naoya Sumi
Journal: Proc. Amer. Math. Soc. 127 (1999), 915-924
MSC (1991): Primary 58F12
MathSciNet review: 1469436
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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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Additional Information

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Tokyo 192-03, Japan

Keywords: Anosov differentiable map, DA-map, sensitive dependence on initial conditions, topological mixing, transversal homoclinic point
Received by editor(s): November 26, 1996
Received by editor(s) in revised form: June 26, 1997
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society