Abstract 𝜔-limit sets, chain recurrent sets, and basic sets for flows

Franke, John E.; Selgrade, James F.

doi:10.1090/S0002-9939-1976-0423423-X

Abstract $\omega$-limit sets, chain recurrent sets, and basic sets for flows
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by John E. Franke and James F. Selgrade

Proc. Amer. Math. Soc. 60 (1976), 309-316

DOI: https://doi.org/10.1090/S0002-9939-1976-0423423-X

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Abstract:

An abstract $\omega$-limit set for a flow is an invariant set which is conjugate to the $\omega$-limit set of a point. This paper shows that an abstract $\omega$-limit set is precisely a connected, chain recurrent set. In fact, an abstract $\omega$-limit set which is a subset of a hyperbolic invariant set is the $\omega$-limit set of a nearby heteroclinic point. This leads to the result that a basic set is a hyperbolic, compact, invariant set which is chain recurrent, connected, and has local product structure.

References

Rufus Bowen, $\omega$-limit sets for axiom $\textrm {A}$ diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333–339. MR 413181, DOI 10.1016/0022-0396(75)90065-0

The gradient structure of a flow

Hyperbolicity and chain recurrence

M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, DOI 10.1007/BF01404552
Charles Pugh and Michael Shub, The $\Omega$-stability theorem for flows, Invent. Math. 11 (1970), 150–158. MR 287579, DOI 10.1007/BF01404608
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1