Lyapunov stability of $\omega$-limit sets

Carlos Arnoldo Morales; M. J. Pacifico

doi:10.3934/dcds.2002.8.671

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2002, Volume 8, Issue 3: 671-674. Doi: 10.3934/dcds.2002.8.671

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

Received: April 2001

Revised: July 2001

Published: July 2002

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Abstract

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].

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

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