Hyperbolic Birkhoff centers

Author:
I. P. Malta

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

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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.

**[1]**R. Bowen,*Periodic points and measures for Axiom*A*diffeomorphisms*, Trans. Amer. Math. Soc.**154**(1971), 377-397. MR**0282372 (43:8084)****[2]**T. Cherry,*Analytic quasi-periodic curves of discontinuous type on a torus*, Proc. London Math. Soc.**44**(1938), 175-215.**[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. 133-163. MR**0271991 (42:6872)****[4]**M. Hirsch, J. Palis, C. Pugh and M. Shub,*Neighborhoods of hyperbolic sets*, Invent. Math.**9**(1969/70), 121-134. MR**0262627 (41:7232)****[5]**S. Newhouse,*Hyperbolic limit sets*, Trans. Amer. Math. Soc.**167**(1972), 125-150. 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. 221-222. MR**0270387 (42:5276)****[8]**-,*On Morse-Smale dynamical systems*, Topology**8**(1968), 385-404. MR**0246316 (39:7620)****[9]**S. Smale,*The*-*stability theorem*, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 289-297. 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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DOI:
https://doi.org/10.1090/S0002-9947-1980-0583851-4

Article copyright:
© Copyright 1980
American Mathematical Society