Generic homeomorphisms have no smallest attractors

Hurley, Mike

doi:10.1090/S0002-9939-1995-1273500-0

Generic homeomorphisms have no smallest attractors
HTML articles powered by AMS MathViewer

by Mike Hurley

Proc. Amer. Math. Soc. 123 (1995), 1277-1280

DOI: https://doi.org/10.1090/S0002-9939-1995-1273500-0

PDF | Request permission

Abstract:

We show that if M is a compact manifold then there is a residual subset $\mathcal {R}$ of the set of homeomorphisms on M with the property that if $f \in \mathcal {R}$ then f has no smallest attractor (that is, an attractor with the property that none of its proper subsets is also an attractor). Part of the motivation for this result comes from portions of a recent paper by Lewowicz and Tolosa that deal with properties of smallest attractors of generic homeomorphisms.

References

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
E. M. Coven, J. Madden, and Z. Nitecki, A note on generic properties of continuous maps, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 97–101. MR 670076
Robert Easton, Isolating blocks and epsilon chains for maps, Phys. D 39 (1989), no. 1, 95–110. MR 1021184, DOI 10.1016/0167-2789(89)90041-9
M. Hurley, Bifurcation and chain recurrence, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 231–240. MR 742225, DOI 10.1017/S0143385700001942
Mike Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 709–729. MR 1145617, DOI 10.1017/S014338570000643X
J. Lewowicz and J. Tolosa, On stable and unstable sets, Dynamical systems (Santiago, 1990) Pitman Res. Notes Math. Ser., vol. 285, Longman Sci. Tech., Harlow, 1993, pp. 117–130. MR 1213944
Kenzi Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc. 110 (1990), no. 1, 281–284. MR 1009998, DOI 10.1090/S0002-9939-1990-1009998-8
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
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