## Quasi-Anosov diffeomorphisms and hyperbolic manifolds

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- by Ricardo Mañé
- Trans. Amer. Math. Soc.
**229**(1977), 351-370 - DOI: https://doi.org/10.1090/S0002-9947-1977-0482849-4
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## Abstract:

Let*f*be a diffeomorphism of a smooth manifold

*N*and $M \subset N$ a compact boundaryless submanifold such that it is a hyperbolic set for

*f*. The diffeomorphism

*f*/

*M*is characterized and it is proved that it is Anosov if and only if

*M*is an invariant isolated set of

*f*(i.e. the maximal invariant subset of some compact neighborhood). Isomorphisms of vector bundles with the property that the zero section is an isolated subset are studied proving that they can be embedded in hyperbolic vector bundle isomorphisms.

## References

- Morris W. Hirsch and Charles C. Pugh,
*Stable manifolds and hyperbolic sets*, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR**0271991** - Morris W. Hirsch,
*On invariant subsets of hyperbolic sets*, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 126–135. MR**0264684** - Ricardo Mañé,
*Expansive diffeomorphisms*, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 162–174. MR**0650658** - John Franks and Clark Robinson,
*A quasi-Anosov diffeomorphism that is not Anosov*, Trans. Amer. Math. Soc.**223**(1976), 267–278. MR**423420**, DOI 10.1090/S0002-9947-1976-0423420-9 - James F. Selgrade,
*Isolated invariant sets for flows on vector bundles*, Trans. Amer. Math. Soc.**203**(1975), 359–390. MR**368080**, DOI 10.1090/S0002-9947-1975-0368080-X - Robert J. Sacker and George R. Sell,
*A note on Anosov diffeomorphisms*, Bull. Amer. Math. Soc.**80**(1974), 278–280. MR**331432**, DOI 10.1090/S0002-9904-1974-13460-9 - Morris W. Hirsch,
*On invariant subsets of hyperbolic sets*, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 126–135. MR**0264684** - John M. Franks,
*Invariant sets of hyperbolic toral automorphisms*, Amer. J. Math.**99**(1977), no. 5, 1089–1095. MR**482846**, DOI 10.2307/2374001 - S. Smale,
*The $\Omega$-stability theorem*, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR**0271971** - Sheldon E. Newhouse,
*Hyperbolic limit sets*, Trans. Amer. Math. Soc.**167**(1972), 125–150. MR**295388**, DOI 10.1090/S0002-9947-1972-0295388-6 - M. Hirsch, J. Palis, C. Pugh, and M. Shub,
*Neighborhoods of hyperbolic sets*, Invent. Math.**9**(1969/70), 121–134. MR**262627**, DOI 10.1007/BF01404552
R. Mañé, - R. Clark Robinson,
*$C^{r}$ structural stability implies Kupka-Smale*, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 443–449. MR**0334282**

*Variedades invariantes*, Tese de doutoramento, Instituto Mat. Pura e Aplicada, Brazil, 1973.

## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**229**(1977), 351-370 - MSC: Primary 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1977-0482849-4
- MathSciNet review: 0482849