Attractors: persistence, and density of their basins

Author:
Mike Hurley

Journal:
Trans. Amer. Math. Soc. **269** (1982), 247-271

MSC:
Primary 58F12; Secondary 54H20, 58F10

DOI:
https://doi.org/10.1090/S0002-9947-1982-0637037-7

MathSciNet review:
637037

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors. A quasi-attractor is any nonempty intersection of attractors. It is shown that quasi-attractors other than attractors occur for a large set of flows. It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows. Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor". Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.

**[B1]**Rufus Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****[BF]**Rufus Bowen and John Franks,*The periodic points of maps of the disk and the interval*, Topology**15**(1976), no. 4, 337–342. MR**431282**, https://doi.org/10.1016/0040-9383(76)90026-4**[Bo]**H. G. Bothe,*A modification of the Kupka-Smale theorem and smooth invariant manifolds of dynamical systems*, Math. Nachr.**89**(1979), 25–42. MR**546870**, https://doi.org/10.1002/mana.19790890105**[BR]**Rufus Bowen and David Ruelle,*The ergodic theory of Axiom A flows*, Invent. Math.**29**(1975), no. 3, 181–202. MR**380889**, https://doi.org/10.1007/BF01389848**[C]**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****[FY]**J. Franks and L. S. Young, preprint, Northwestern Univ., 1980.**[G]**J. E. Marsden and M. McCracken,*The Hopf bifurcation and its applications*, 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; Applied Mathematical Sciences, Vol. 19. MR**0494309****[GA]***Global analysis*, Proceedings of Symposia in Pure Mathematics, Vols. XIV-XVI. Edited by Shiing-shen Chern and Stephen Smale, American Mathematical Society, Providence, R.I., 1970. MR**0263081****[H]**Philip Hartman,*Ordinary differential equations*, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR**0344555****[Hi]**Morris W. Hirsch,*Differential topology*, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR**0448362****[HPS]**M. W. Hirsch, C. C. Pugh, and M. Shub,*Invariant manifolds*, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR**0501173****[K]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[Lo]**Artur Oscar Lopes,*Structural stability and hyperbolic attractors*, Trans. Amer. Math. Soc.**252**(1979), 205–219. MR**534118**, https://doi.org/10.1090/S0002-9947-1979-0534118-3**[M]**Anthony Manning,*Topological entropy and the first homology group*, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 185–190. Lecture Notes in Math., Vol. 468. MR**0650661****[N]**Zbigniew Nitecki,*Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms*, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR**0649788****[NS]**Z. Nitecki and M. Shub,*Filtrations, decompositions, and explosions*, Amer. J. Math.**97**(1975), no. 4, 1029–1047. MR**394762**, https://doi.org/10.2307/2373686**[Na]**Sam B. Nadler Jr.,*Hyperspaces of sets*, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions; Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49. MR**0500811****[Ne1]**Sheldon E. Newhouse,*Nondensity of axiom 𝐴(𝑎) on 𝑆²*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR**0277005****[Ne2]**Sheldon E. Newhouse,*Diffeomorphisms with infinitely many sinks*, Topology**13**(1974), 9–18. MR**339291**, https://doi.org/10.1016/0040-9383(74)90034-2**[O]**M. M. C. de Oliveira,*𝐶⁰-density of structurally stable vector fields*, Bull. Amer. Math. Soc.**82**(1976), no. 5, 786. MR**420716**, https://doi.org/10.1090/S0002-9904-1976-14165-1**[P]**Charles C. Pugh,*An improved closing lemma and a general density theorem*, Amer. J. Math.**89**(1967), 1010–1021. MR**226670**, https://doi.org/10.2307/2373414**[PP]**J. Palis, C. Pugh, M. Shub, and D. Sullivan,*Genericity theorems in topological dynamics*, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 241–250. Lecture Notes in Math., Vol. 468. MR**0650665****[S]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**228014**, https://doi.org/10.1090/S0002-9904-1967-11798-1**[Sh1]**Michael Shub,*Structurally stable diffeomorphisms are dense*, Bull. Amer. Math. Soc.**78**(1972), 817–818. MR**307278**, https://doi.org/10.1090/S0002-9904-1972-13047-7**[Sh2]**M. Shub,*Dynamical systems, filtrations and entropy*, Bull. Amer. Math. Soc.**80**(1974), 27–41. MR**334284**, https://doi.org/10.1090/S0002-9904-1974-13344-6**[Sh3]**M. Shub,*Stability and genericity for diffeomorphisms*, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 493–514. MR**0331431****[SS]**M. Shub and S. Smale,*Beyond hyperbolicity*, Ann. of Math. (2)**96**(1972), 587–591. MR**312001**, https://doi.org/10.2307/1970826**[T]**René Thom,*Stabilité structurelle et morphogénèse*, W. A. Benjamin, Inc., Reading, Mass., 1972 (French). Essai d’une théorie générale des modèles; Mathematical Physics Monograph Series. MR**0488155****[Ta1]**Floris Takens,*On Zeeman’s tolerance stability conjecture*, Manifolds—Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 209–219. MR**0279790****[Ta2]**Floris Takens,*Tolerance stability*, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 293–304. Lecture Notes in Math., Vol. 468. MR**0650298****[Wi1]**R. F. Williams,*The “𝐷𝐴” maps of Smale and structural stability*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 329–334. MR**0264705****[Wi2]**R. F. Williams,*The structure of Lorenz attractors*, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Springer, Berlin, 1977, pp. 94–112. Lecture Notes in Math., Vol. 615. MR**0461581****[Wi3]**-,*The structure of Lorenz attractors*, preprint, Northwestern Univ., 1978.**[Z]**E. C. Zeeman,*Morse inequalities for diffeomorphisms with shoes and flows with solenoids*, Dynamical Systems, Lecture Notes in Math., vol. 468, Springer-Verlag, New York, 1975, pp. 44-47.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F12,
54H20,
58F10

Retrieve articles in all journals with MSC: 58F12, 54H20, 58F10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0637037-7

Keywords:
Attractor,
quasi-attractor,
chain recurrence,
chain transitivity

Article copyright:
© Copyright 1982
American Mathematical Society