Timeline for $\omega$-limits of $1$-dimensional dynamical systems
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2017 at 21:43 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 1, 2017 at 20:58 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| Dec 3, 2016 at 17:31 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 3, 2016 at 15:41 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 4, 2016 at 18:30 | comment | added | Will Brian | Any subset of $[0,1]$ can be obtained as an $\omega$-limit set coming from a continuous map on $[0,1]^2$. So bumping up the dimension let's you get whatever you want. (In fact, one can even use a one-dimensional (but not connected) subspace of $[0,1]^2$ -- so you can even keep one-dimensionality, provided you're willing to sacrifice working with manifolds.) But I don't know what is possible for continuous functions or for homeomorphisms of $[0,1]$ or $S^1$ to itself. Interesting question! | |
| Oct 4, 2016 at 14:19 | comment | added | Ian Morris | @Asaf: for 4 your limit set will also include preperiodic points of the form $00000\ldots 00000\overline{10}$. | |
| Oct 4, 2016 at 14:16 | answer | added | Ian Morris | timeline score: 1 | |
| Oct 1, 2016 at 16:57 | comment | added | Asaf | For $5$ I think it's possible to use the strategy of $4$, order somehow the infinitely (countably-)many periodic orbits for $\times 2$ (they are contained inside discrete subgroups of the torus), and then just mimic the construction I gave, you will "see" much of every orbit for many times hence it will be inside your $\omega$-limit. | |
| Oct 1, 2016 at 16:54 | comment | added | Asaf | For 2 say, assuming your map is ergodic (wrt to Lebesgue), as you will contain an open set, this open set would contain a generic point, and the orbit of a generic point is dense (and even equidistributed). | |
| Oct 1, 2016 at 16:53 | comment | added | Asaf | For an example to 4, think about $\times 2$ action on the circle (I think about it as a Bernoulli system). $0$ is a fixed point, and $\{1/3,2/3\}$ is a period, so just track the trajectory along the period, and sometimes just stick a large bunch of $0$'s into your expansion, you'll get a point which for many times close to the period, but for many other times close to $0$, hence you can get such a $\omega$-limit. | |
| Oct 1, 2016 at 16:51 | comment | added | Asaf | Kronecker systems are semisimple - they break down into disjoint union of minimal subsystems, and as every point in a minimal subsystem is dense in the orbit closure and uniformly recurrent, it is easy to classify its w-limits. Thing start to become interesting when you're not inside a minimal system (or semisimple). | |
| Oct 1, 2016 at 15:02 | history | asked | Selim G | CC BY-SA 3.0 |