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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A class of differentiable toral maps which are topologically mixing
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by Naoya Sumi PDF
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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Additional Information
  • Naoya Sumi
  • Affiliation: Department of Mathematics, Tokyo Metropolitan University, Tokyo 192-03, Japan
  • MR Author ID: 610209
  • Email: sumi@math.metro-u.ac.jp
  • Received by editor(s): November 26, 1996
  • Received by editor(s) in revised form: June 26, 1997
  • Communicated by: Mary Rees
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 915-924
  • MSC (1991): Primary 58F12
  • DOI: https://doi.org/10.1090/S0002-9939-99-04608-0
  • MathSciNet review: 1469436