Hyperbolic Birkhoff centers
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- by I. P. Malta PDF
- Trans. Amer. Math. Soc. 262 (1980), 181-193 Request permission
Abstract:
The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, $c(f)$, is hyperbolic and has no cycles, then f satisfies Axiom A and is $\Omega$-stable. To obtain a filtration for $c(f)$, the concept of an isolated set for a homeomorphism of a compact metric space is introduced. As a partial converse it is proved that if $c(f)$ is hyperbolic and f is $\Omega$-stable, then $c(f)$ has the no cycle property. A characterization of $\Omega$-stability when $c(f)$ is finite is also given.References
- Rufus Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397. MR 282372, DOI 10.1090/S0002-9947-1971-0282372-0 T. Cherry, Analytic quasi-periodic curves of discontinuous type on a torus, Proc. London Math. Soc. 44 (1938), 175-215.
- 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
- 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
- Sheldon E. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125–150. MR 295388, DOI 10.1090/S0002-9947-1972-0295388-6 —, Lectures on dynamical systems (C.I.M.E. Summer Session in Dynamical Systems, Bressanone, Italy, June 1978).
- J. Palis, A note on $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 221–222. MR 0270387
- J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, DOI 10.1016/0040-9383(69)90024-X
- S. Smale, The $\Omega$-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 —, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J., 1975.
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 262 (1980), 181-193
- MSC: Primary 58F15; Secondary 58F20
- DOI: https://doi.org/10.1090/S0002-9947-1980-0583851-4
- MathSciNet review: 583851