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The stability theorems for discrete dynamical systems on two-dimensional manifolds

Published online by Cambridge University Press:  22 January 2016

Atsuro Sannami
Atsuro Sannami*
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo, Japan
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One of the basic problems in the theory of dynamical systems is the characterization of stable systems.

Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 90 , June 1983 , pp. 1 - 55
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1983

References

[ 1 ] Franks, J., Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc, 158 (1971), 301–308.Google Scholar
[ 2 ] Hirsch, M., Palis, J., Pugh, C. and Shub, M., Neighborhoods of hyperbolic sets, Invent. Math., 9 (1970), 121–134.Google Scholar
[ 3 ] Hirsch, M. and Pugh, C., Stable manifolds and hyperbolic sets, Proc. of Sympo. Pure Math. (Global Analysis) XIV, Amer. Math. Soc, (1970), 133–163.Google Scholar
[ 4 ] Hirsch, M., Pugh, C. and Shub, M., Invariant Manifolds, Lee. Notes in Math. 583, Springer.Google Scholar
[ 5 ] Mañé, R., Expansive diffeomorphisms. Dynamical Systems, Warwick, 1974. Lee. Notes in Math. 468, Springer.Google Scholar
[ 6 ] Mañé, R., Contributions to the stability conjecture, Topology, 17 (1978), 383–396.Google Scholar
[ 7 ] Mather, J., Characterization of Anosov diffeomorphisms, Indag. Math., 30 (1968), 479–483.Google Scholar
[ 8 ] Palis, J., A note on Ω-stability, Proc. of Sympo. Pure Math. (Global Analysis), XIV, Amer. Math. Soc, 221–222.Google Scholar
[ 9 ] Palis, J. and Smale, S., Structural stability theorems, Proc. of Sympo. Pure Math. (Global Analysis), XIV, Amer. Math. Soc, (1970), 223–231.Google Scholar
[10] Pliss, V. A., The location of separatrices of periodic saddle-point motion of systems of second-order differential equations, Differential Equations, 7 (1971), 906–927.Google Scholar
[11] Pliss, V. A., A hypothesis due to Smale, Differential Equations, 8 (1972), 203–214.Google Scholar
[12] Pugh, C. and Robinson, C., The C1 Closing Lemma, Including Hamiltonians, Preprint.Google Scholar
[13] Robbin, J., A structural stability theorem. Ann. of Math., 94 (1971), 447–493.Google Scholar
[14] Robinson, C., Cr structural stability implies Kupka-Smale, Dynamical systems-Salvador, M. Peixoto (Editor), Academic Press, 1973.Google Scholar
[15] Robinson, C., Structural stability of C1 diffeomorphisms, J. Differential Equations, 22 (1976), 28–73.Google Scholar
[16] Robinson, C., Stability, measure zero, and dimension two implies hyperbolicity, Preprint.Google Scholar
[17] Smale, S., Differentiate dynamical systems, Bull. Amer. Math. Soc, 73 (1967), 747–817.Google Scholar
[18] Smale, S., The Ω-stability theorem, Proc. of Sympo. Pure Math. (Global Analysis), 14. Amer. Math. Soc, (1970), 289–297.Google Scholar