Hyperbolic Birkhoff centers
Author:
I. P. Malta
Journal:
Trans. Amer. Math. Soc. 262 (1980), 181193
MSC:
Primary 58F15; Secondary 58F20
MathSciNet review:
583851
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Abstract: The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, , is hyperbolic and has no cycles, then f satisfies Axiom A and is stable. To obtain a filtration for , 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 is hyperbolic and f is stable, then has the no cycle property. A characterization of stability when is finite is also given.
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 R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377397. MR 0282372 (43:8084)
 [2]
 T. Cherry, Analytic quasiperiodic curves of discontinuous type on a torus, Proc. London Math. Soc. 44 (1938), 175215.
 [3]
 M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 133163. MR 0271991 (42:6872)
 [4]
 M. Hirsch, J. Palis, C. Pugh and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121134. MR 0262627 (41:7232)
 [5]
 S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125150. MR 0295388 (45:4454)
 [6]
 , Lectures on dynamical systems (C.I.M.E. Summer Session in Dynamical Systems, Bressanone, Italy, June 1978).
 [7]
 J. Palis, A note on stability, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 221222. MR 0270387 (42:5276)
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 , On MorseSmale dynamical systems, Topology 8 (1968), 385404. MR 0246316 (39:7620)
 [9]
 S. Smale, The stability theorem, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 289297. MR 0271971 (42:6852)
 [10]
 , Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J., 1975.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005838514
PII:
S 00029947(1980)05838514
Article copyright:
© Copyright 1980
American Mathematical Society
