Timeline for About dense orbits on dynamical systems
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 17, 2010 at 9:07 | comment | added | Kaminoite | I see.. The problem that I found is that it not assures that $p$ is in the intersection. Nevertheless, as you said, the Baire's theorem asserts that this intersection is dense but, by Borel-Cantelli lemma, that its measure is $0$. | |
| Apr 16, 2010 at 20:19 | comment | added | Ian Morris | Oops, I should have said for $\delta$ tending to infinity, not zero. I don't think that there's a problem with Gerald's proof: given any dense orbit $(x_n)$ and real number $\delta>0$, Baire's theorem shows that there exists $p \in \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty B(x_n,e^{-n\delta})$, and by unravelling the meaning of this statement we find that this $p$ satisfies $d(x_n,p)<e^{-n\delta})$ for infinitely many $n$. | |
| Apr 16, 2010 at 17:46 | vote | accept | Kaminoite | ||
| Apr 17, 2010 at 9:08 | |||||
| Apr 16, 2010 at 17:46 | vote | accept | Kaminoite | ||
| Apr 16, 2010 at 17:46 | |||||
| Apr 16, 2010 at 17:25 | comment | added | Kaminoite | Ian, I think that Gerald's proof is not correct? Am I wrong? | |
| Apr 16, 2010 at 16:52 | comment | added | Ian Morris | Nice! We could even take a further intersection over a sequence of $\delta$'s tending to zero, proving the existence of points which work simultaneously for all $\delta$. | |
| Apr 8, 2010 at 19:55 | comment | added | Kaminoite | This does not mean that $p$ is in the intersection but in its closure. | |
| Apr 8, 2010 at 16:46 | history | edited | Gerald Edgar | CC BY-SA 2.5 |
added 137 characters in body; deleted 2 characters in body
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| Apr 8, 2010 at 16:40 | history | answered | Gerald Edgar | CC BY-SA 2.5 |