## Fixed points of topologically stable flows

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- by Mike Hurley
- Trans. Amer. Math. Soc.
**294**(1986), 625-633 - DOI: https://doi.org/10.1090/S0002-9947-1986-0825726-4
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## Abstract:

This paper concerns certain necessary conditions for a flow to be topologically stable (in the sense of P. Walters). In particular, it is shown that under fairly general conditions one can conclude that a topologically stable flow has a finite number of fixed points, and each of these is isolated in the chain recurrent set of the flow.## References

- Rufus Bowen,
*Periodic orbits for hyperbolic flows*, Amer. J. Math.**94**(1972), 1–30. MR**298700**, DOI 10.2307/2373590 - Rufus Bowen,
*One-dimensional hyperbolic sets for flows*, J. Differential Equations**12**(1972), 173–179. MR**336762**, DOI 10.1016/0022-0396(72)90012-5 - Rufus Bowen,
*Symbolic dynamics for hyperbolic flows*, Amer. J. Math.**95**(1973), 429–460. MR**339281**, DOI 10.2307/2373793 - Rufus Bowen,
*On Axiom A diffeomorphisms*, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR**0482842** - Charles Conley,
*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133**
—, - Philip Fleming and Mike Hurley,
*A converse topological stability theorem for flows on surfaces*, J. Differential Equations**53**(1984), no. 2, 172–191. MR**748238**, DOI 10.1016/0022-0396(84)90038-X - John M. Franks,
*Homology and dynamical systems*, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C. by the American Mathematical Society, Providence, R.I., 1982. MR**669378** - J. E. Marsden and M. McCracken,
*The Hopf bifurcation and its applications*, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale. MR**0494309** - John Guckenheimer and R. F. Williams,
*Structural stability of Lorenz attractors*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 59–72. MR**556582** - Jack K. Hale,
*Ordinary differential equations*, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR**0419901** - Mike Hurley,
*Attractors: persistence, and density of their basins*, Trans. Amer. Math. Soc.**269**(1982), no. 1, 247–271. MR**637037**, DOI 10.1090/S0002-9947-1982-0637037-7 - Mike Hurley,
*Consequences of topological stability*, J. Differential Equations**54**(1984), no. 1, 60–72. MR**756545**, DOI 10.1016/0022-0396(84)90142-6 - Mike Hurley,
*Combined structural and topological stability are equivalent to Axiom A and the strong transversality condition*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 81–88. MR**758895**, DOI 10.1017/S0143385700002285 - Kazuhisa Kato and Akihiko Morimoto,
*Topological stability of Anosov flows and their centralizers*, Topology**12**(1973), 255–273. MR**326779**, DOI 10.1016/0040-9383(73)90012-8 - Kazuhisa Kato and Akihiko Morimoto,
*Topological $\Omega$-stability of Axiom A flows with no $\Omega$-explosions*, J. Differential Equations**34**(1979), no. 3, 464–481. MR**555322**, DOI 10.1016/0022-0396(79)90031-7 - M. M. C. de Oliveira,
*$C^{0}$-density of structurally stable vector fields*, Bull. Amer. Math. Soc.**82**(1976), no. 5, 786. MR**420716**, DOI 10.1090/S0002-9904-1976-14165-1 - Jacob Palis Jr. and Welington de Melo,
*Geometric theory of dynamical systems*, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR**669541** - Clark Robinson,
*Stability theorems and hyperbolicity in dynamical systems*, Rocky Mountain J. Math.**7**(1977), no. 3, 425–437. MR**494300**, DOI 10.1216/RMJ-1977-7-3-425 - S. Smale,
*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**228014**, DOI 10.1090/S0002-9904-1967-11798-1 - Colin Sparrow,
*The Lorenz equations: bifurcations, chaos, and strange attractors*, Applied Mathematical Sciences, vol. 41, Springer-Verlag, New York-Berlin, 1982. MR**681294** - Peter Walters,
*Anosov diffeomorphisms are topologically stable*, Topology**9**(1970), 71–78. MR**254862**, DOI 10.1016/0040-9383(70)90051-0 - R. F. Williams,
*The structure of Lorenz attractors*, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Lecture Notes in Math., Vol. 615, Springer, Berlin, 1977, pp. 94–112. MR**0461581** - John Guckenheimer and R. F. Williams,
*Structural stability of Lorenz attractors*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 59–72. MR**556582**
E. C. Zeeman,

*The gradient structure of a flow*, I.B.M. Res. RC 3932. Yorktown Heights, N.Y., 1972.

*Morse inequalities for diffeomorphisms with shoes and flows with solenoids*, Dynamical Systems-Warwick 1974, Lecture Notes in Math., vol. 468, Springer-Verlag, New York, 1975, pp. 44-47.

## Bibliographic Information

- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**294**(1986), 625-633 - MSC: Primary 58F25
- DOI: https://doi.org/10.1090/S0002-9947-1986-0825726-4
- MathSciNet review: 825726