Attractors: persistence, and density of their basins

Hurley, Mike

doi:10.1090/S0002-9947-1982-0637037-7

Attractors: persistence, and density of their basins
HTML articles powered by AMS MathViewer

by Mike Hurley PDF

Trans. Amer. Math. Soc. 269 (1982), 247-271 Request permission

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.

References

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
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, DOI 10.1016/0040-9383(76)90026-4
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, DOI 10.1002/mana.19790890105
Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
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
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
Shiing-shen Chern and Stephen Smale (eds.), Global analysis, Proceedings of Symposia in Pure Mathematics, XIV-XVI, American Mathematical Society, Providence, R.I., 1970. MR 0263081
Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
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. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Artur Oscar Lopes, Structural stability and hyperbolic attractors, Trans. Amer. Math. Soc. 252 (1979), 205–219. MR 534118, DOI 10.1090/S0002-9947-1979-0534118-3
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), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 185–190. MR 0650661
Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1975), no. 4, 1029–1047. MR 394762, DOI 10.2307/2373686
Sam B. Nadler Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions. MR 0500811
Sheldon E. Newhouse, Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR 0277005
Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR 339291, DOI 10.1016/0040-9383(74)90034-2
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
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
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), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 241–250. MR 0650665
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Michael Shub, Structurally stable diffeomorphisms are dense, Bull. Amer. Math. Soc. 78 (1972), 817–818. MR 307278, DOI 10.1090/S0002-9904-1972-13047-7
M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc. 80 (1974), 27–41. MR 334284, DOI 10.1090/S0002-9904-1974-13344-6
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
M. Shub and S. Smale, Beyond hyperbolicity, Ann. of Math. (2) 96 (1972), 587–591. MR 312001, DOI 10.2307/1970826
René Thom, Stabilité structurelle et morphogénèse, Mathematical Physics Monograph Series, W. A. Benjamin, Inc., Reading, Mass., 1972 (French). Essai d’une théorie générale des modèles. MR 0488155
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
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), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 293–304. MR 0650298
R. F. Williams, The $“\textrm {DA}''$ 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
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

The structure of Lorenz attractors

Morse inequalities for diffeomorphisms with shoes and flows with solenoids