Timeline for Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 28, 2023 at 4:02 | comment | added | Alapan Das | @StevenStadnicki Oh, I see. Thank you for clearing my doubt. | |
| Feb 27, 2023 at 20:15 | comment | added | Steven Stadnicki | @AlapanDas Even if they do converge to such a solution, that doesn't mean that the sequence 'leads to' periodicity in the sense OP means — they're specifically asking (AFAICT) if 1/3 is a preperiodic point. | |
| Feb 27, 2023 at 18:05 | comment | added | Alapan Das | I may be navie, but how to ensure that $x_{lr}, l=1,2....$ for some $r$ (period) don't tend to an irrational solution of $f^{(r)}(x)=x$ where $f(x)=4x(1-x)$? | |
| Feb 27, 2023 at 16:02 | vote | accept | Vincent Granville | ||
| Feb 27, 2023 at 14:42 | comment | added | Noam D. Elkies | That's all correct; it can also be explained in more elementary (and general) fashion: if x is rational with odd denominator q, then 4x(1-x) has denominator q^2, so by induction the n-th iterate has denominator q^(2^n), and in particular x is never periodic once q>1. | |
| S Feb 27, 2023 at 13:26 | review | First answers | |||
| Feb 27, 2023 at 13:28 | |||||
| S Feb 27, 2023 at 13:26 | history | edited | Bill Bradley | CC BY-SA 4.0 |
Minor typo.
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| Feb 27, 2023 at 12:13 | history | edited | user500150 | CC BY-SA 4.0 |
fixed indexing error
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| Feb 27, 2023 at 12:10 | history | edited | user500150 | CC BY-SA 4.0 |
edited body
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| Feb 27, 2023 at 12:09 | history | edited | user500150 | CC BY-SA 4.0 |
added 1 character in body
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| S Feb 27, 2023 at 12:07 | review | First answers | |||
| Feb 27, 2023 at 13:06 | |||||
| S Feb 27, 2023 at 12:07 | history | answered | user500150 | CC BY-SA 4.0 |