I can not understand the difference between omega limit of a point and its stable manifold? Anybody can give me some example please?
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1$\begingroup$ This isn't really research-level and you may have better luck on math.stackexchange.com (as per the FAQ here), but FWIW you may perhaps gain some insight by sonsidering the example $f\colon \mathbb{R}^2\to\mathbb{R}^2$ given by $f(x,y) = (2x, y/2)$, and using the definitions to work out the omega limit set and stable manifold of $(0,0)$. Then try $(0,1)$ and $(1,0)$. $\endgroup$– Vaughn ClimenhagaAug 29, 2012 at 15:40
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$\begingroup$ As @Vaughn Climenhaga said, this question is not research level. But maybe a quick answer is that if $x$ is in the omega limit set of $p$, then $p$ is in the stable manifold of $x$ and vice versa. $\endgroup$– user7807Aug 31, 2012 at 22:40
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