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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diffeomorphisms approximated by Anosov on the
2-torus and their SBR measures


Author: Naoya Sumi
Journal: Trans. Amer. Math. Soc. 351 (1999), 3373-3385
MSC (1991): Primary 58F11, 58F12, 58F15
Published electronically: April 8, 1999
MathSciNet review: 1637098
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Abstract: We consider the $C^{2}$ set of $C^{2}$ diffeomorphisms of the 2-torus $\mathbb{T}^{2}$, provided the conditions that the tangent bundle splits into the directed sum $T\mathbb{T}^{2}=E^{s}\oplus E^{u}$ of $Df$-invariant subbundles $E^{s}$, $E^{u}$ and there is $0<\lambda <1$ such that $\Vert Df|_{E^{s}}\Vert <\lambda $ and $\Vert Df|_{E^{u}}\Vert \ge 1 $. Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the $C^{2}$ set has no SBR measures. This is related to a result of Hu-Young.


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Additional Information

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-03, Japan
Email: sumi@math.metro-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02426-5
PII: S 0002-9947(99)02426-5
Keywords: Anosov diffeomorphism, SBR measure
Received by editor(s): February 10, 1997
Published electronically: April 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society