Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-99-04608-0
MathSciNet review: 1469436
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\mathbb{T}^{n}$ ($n\ge 4$).


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58F12

Retrieve articles in all journals with MSC (1991): 58F12


Additional Information

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Tokyo 192-03, Japan
Email: sumi@math.metro-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-99-04608-0
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