Lyapunov stability of \omega-limit sets

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July 2002, 8(3): 671-674. doi: 10.3934/dcds.2002.8.671

Lyapunov stability of $\omega$-limit sets

Carlos Arnoldo Morales ^1, and M. J. Pacifico ^2,

1.	Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil
2.	Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil

Received April 2001 Revised July 2001 Published April 2002

Abstract
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We prove that there is a residual subset $C$, in the space of all $\mathcal C^1$ vector fields of a closed $n$-manifold $M$, such that for every $X \in \mathcal R$ the set of points in $M$ with Lyapunov stable $\omega$-limit set is residual in $M$. This improves a result in Arnaud [1] and gives a partial solution to a conjecture in Hurley [8].

Keywords: nonwandering set., Lyapunov stable, $\omega$-limit set.

Mathematics Subject Classification: Primary: 37Dxx, 37Bxx, 34Bx.

Citation: Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

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