Hyperbolic limit sets

Newhouse, Sheldon E.

doi:10.1090/S0002-9947-1972-0295388-6

Hyperbolic limit sets

Author: Sheldon E. Newhouse
Journal: Trans. Amer. Math. Soc. 167 (1972), 125-150
MSC: Primary 58F15; Secondary 34C35
DOI: https://doi.org/10.1090/S0002-9947-1972-0295388-6
MathSciNet review: 0295388
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set ${L^ - }(f)$ of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in ${L^ - }(f)$ . A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if ${L^ - }(f)$ is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is $\Omega$ -stable. Examples are given where ${L^ - }(f)$ is hyperbolic, there are cycles, and f fails to satisfy Axiom A.

[1] George D. Birkhoff, Dynamical systems, With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. MR 0209095
[2] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
[3] M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, https://doi.org/10.1007/BF01404552
[4] J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, https://doi.org/10.1016/0040-9383(69)90024-X
[5] -, A note on $\Omega$ -stability, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970. MR 41 #7686.
[6] J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
[7] Charles Pugh and Michael Shub, The Ω-stability theorem for flows, Invent. Math. 11 (1970), 150–158. MR 287579, https://doi.org/10.1007/BF01404608
[8] J. W. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447–493. MR 287580, https://doi.org/10.2307/1970766
[9] Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020
[10] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, https://doi.org/10.1090/S0002-9904-1967-11798-1
[11] S. Smale, The Ω-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, r.I., 1970, pp. 289–297. MR 0271971
[12] Zbigniew Nitecki, On semi-stability for diffeomorphisms, Invent. Math. 14 (1971), 83–122. MR 293671, https://doi.org/10.1007/BF01405359
[13] R. Abraham and S. Smale, Nongenericity of Ω-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR 0271986
[14] L. P. Šil′nikov, On the question of the structure of the neighborhood of a homoclinic tube of an invariant torus, Dokl. Akad. Nauk SSSR 180 (1968), 286–289 (Russian). MR 0248861