Timeline for Solve the functional equation $f(4x(1-x))=\sin(\pi f(x))$ to find an invariant measure of a dynamical system $x_{n+1}=\sin(\pi x_{n})$
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 4, 2017 at 12:32 | answer | added | David E Speyer | timeline score: 6 | |
| May 4, 2017 at 11:53 | history | edited | Ian Morris |
edited tags
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| May 3, 2017 at 23:18 | answer | added | Anthony Quas | timeline score: 10 | |
| S May 3, 2017 at 23:15 | history | suggested | jeq | CC BY-SA 3.0 |
Corrected some English typos.
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| May 3, 2017 at 23:00 | review | Suggested edits | |||
| S May 3, 2017 at 23:15 | |||||
| May 3, 2017 at 22:45 | comment | added | user78249 | Not sure if this helps, but letting $h = 4x(1-x)$ and $g(x) = \sin(\pi x)$, the way I usually solve conjugation problems like this is working with a limit. (By conjugation I mean $f$ conjugates between $h$ and $g$.) So for instance, if we let $f(x) = \lim_{n\to\infty} g^{\circ -n}(h^{\circ n}(x))$, then trivially it will satisfy your equation. Sadly, it's tough to prove convergence, and I have no idea if it's injective, and I doubt it would work on the entire interval $(0,1)$. But since both functions are symmetric about $1/2$ and injective there, it may work on $(0,1/2)$. | |
| May 3, 2017 at 22:02 | history | asked | matematicaActiva | CC BY-SA 3.0 |