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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 | References | Similar Articles | Additional Information

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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  • [A] D.V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967), 1-235. MR 39:3527
  • [A-H] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Mathematical Library, North-Holland, 1994. MR 95m:58095
  • [B] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer-Verlag, 1975. MR 56:1364
  • [H] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117-162. MR 91b:58184
  • [H-Y] H. Hu and L.S. Young, Nonexistence of SBR measures for some diffeomorphisms that are `Almost Anosov', Ergod. Th. & Dynam. Sys. 15 (1995), 67-76. MR 95j:58096
  • [H-P] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis, Proc. Sympos. Pure Math. 14, Amer. Math. Soc. (1970), 133-163. MR 42:6872
  • [L] F. Ledrappier, Propriétés ergodiques des measures de Sinai, Publ. Math. I.H.E.S. 59 (1984), 163-188. MR 86f:58092
  • [L-Y] R. Ledrappier and L.S. Young, The metric entropy of diffeomorphisms I, Ann. of Math. 122 (1985), 509-539. MR 87i:58101a
  • [M] R. Mañé, Contributions to the stability conjecture, Topology 17 (1978), 383-396. MR 84b:58061
  • [P] Y.B. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents, Math. USSR Izvestija 10 (1978), 1261-1305. MR 56:16690
  • [R1] J. Robbin, A structural stability theorem, Ann. of Math. 94 (1971), 447-493. MR 44:4783
  • [R2] C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), 425-437. MR 58:13200
  • [R3] V.A. Rohlin, Lectures on the theory of entropy of translations with invariant measures, Russian Math. Surveys 22:5 (1967), 1-54. MR 36:349
  • [S1] Ya.G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 166 (1972), 21-69. MR 53:3265
  • [S2] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37:3598

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Additional Information

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-03, Japan

Keywords: Anosov diffeomorphism, SBR measure
Received by editor(s): February 10, 1997
Published electronically: April 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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