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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**Rufus Bowen,*Periodic points and measures for Axiom 𝐴 diffeomorphisms*, Trans. Amer. Math. Soc.**154**(1971), 377–397. MR**0282372**, https://doi.org/10.1090/S0002-9947-1971-0282372-0**[2]**T. Cherry,*Analytic quasi-periodic curves of discontinuous type on a torus*, Proc. London Math. Soc.**44**(1938), 175-215.**[3]**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****[4]**M. Hirsch, J. Palis, C. Pugh, and M. Shub,*Neighborhoods of hyperbolic sets*, Invent. Math.**9**(1969/1970), 121–134. MR**0262627**, https://doi.org/10.1007/BF01404552**[5]**Sheldon E. Newhouse,*Hyperbolic limit sets*, Trans. Amer. Math. Soc.**167**(1972), 125–150. MR**0295388**, https://doi.org/10.1090/S0002-9947-1972-0295388-6**[6]**-,*Lectures on dynamical systems*(C.I.M.E. Summer Session in Dynamical Systems, Bressanone, Italy, June 1978).**[7]**J. Palis,*A note on Ω-stability*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 221–222. MR**0270387****[8]**J. Palis,*On Morse-Smale dynamical systems*, Topology**8**(1968), 385–404. MR**0246316**, https://doi.org/10.1016/0040-9383(69)90024-X**[9]**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****[10]**-,*Diffeomorphisms with many periodic points*, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J., 1975.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F15,
58F20

Retrieve articles in all journals with MSC: 58F15, 58F20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0583851-4

Article copyright:
© Copyright 1980
American Mathematical Society