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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

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 and Continuous Dynamical Systems, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671
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