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Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of $x_{0}$? Thanks

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You should explain your notation: what do $\omega(x)$ and $\gamma^+(x)$ mean? –  Jaap Eldering Mar 12 at 9:53
    
I think $\gamma^+(x)$ is the forward orbit through $x$ and $\omega(x)$ is the $\omega$-limit set of that orbit. –  Michael Renardy Mar 12 at 14:27
    
Thanks Michael. This notation is commonly used. Thanks –  user31317 Mar 12 at 14:56

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