Timeline for When is convergence transitive?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2010 at 17:24 | comment | added | rpotrie | It is easy to construct an example where $\omega(\omega(X))$ is strictly contained in $\omega(X)$ (consider a rotation of the circle wich attracts a neighborhood in the plane, and multiply the vector field by a bump function which vanishes in a point of the circle). So, if the question is stated as: Does $\omega(X) \subset Y$ and $\omega(Y) \subset Z$ imply $\omega(X) \subset Z$? The answer is no. I guess I had misunderstood the question when I answered it. | |
| Dec 10, 2010 at 17:54 | comment | added | Vincenzo | Still thinking about this, sorry... $\omega(\omega(X))$ is certainly contained in $\omega(X)$, but is it always the case that $\omega(\omega(X))=\omega(X)$? If yes, why? And if no, then your argument only proves that $\omega(X)$ and $Z$ intersect, while I need to argue that $\omega(X)$ is contained in $Z$... | |
| Nov 25, 2010 at 10:13 | comment | added | rpotrie | If you prefer to put the hipotesis that the omega limit of every point of $X$ is contained (instead of intersects) in $Y$ the same argument works. You should look at the kew word omega limit, see en.wikipedia.org/wiki/Limit_set | |
| Nov 25, 2010 at 10:11 | comment | added | rpotrie | Let me try to be more specific: If the equation integrates into a flow (that is, $\varphi_{t+s}(x)=\varphi_t(\varphi_s(x))$) we get that we can define the omega-limit of a point (as the set of points to which the orbit converges) which is an invariant set and the omega limit set of a omega limit set is contained in itself. Consider $x\in X$, then its omega limit intersects $Y$, but since it is invariant and every point of $Y$ has its omega limit intersecting $Z$ we conclude that the omega limit set of $x$ must intersect $Z$ also. | |
| Nov 25, 2010 at 10:04 | comment | added | Vincenzo | Thanks rpotrie. For the (high-dimensional) locally Lipschitz case, do you know if the argument you just sketched is formalized in some standard text, or maybe some standard theorem from which it descends immediately? | |
| Nov 25, 2010 at 9:56 | vote | accept | Vincenzo | ||
| Nov 24, 2010 at 21:43 | history | answered | rpotrie | CC BY-SA 2.5 |