Timeline for Do complex iterates of functions have any meaning?
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 13, 2023 at 21:27 | comment | added | Tom Copeland | The formal group law approach is also used in "Fractional iteration of series and transseries" by Edgar (ams.org/journals/tran/2013-365-11/S0002-9947-2013-05784-4/…). | |
| Sep 7, 2022 at 8:08 | comment | added | Gottfried Helms | thanks,Tom, for that notice. I'll look up your hints to be read; and of course you made me curious about your 80 pages ... | |
| Sep 6, 2022 at 18:39 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added Ward refs
|
| Mar 12, 2022 at 3:12 | comment | added | Tom Copeland | @GottfriedHelms, I provide two simple interpretations--one functional, the other, operational, which have a long history beginning with the foundations of iteration theory--of the original question "Do complex iterates of functions have any meaning?" This is not the venue for a comprehensive discussion of IT, which is quite wide and deep. In fact, I have 80 or so pages of notes I've been sitting on since July that I feel only scratch the surface of the subject. Perhaps Curtright and Zachos, in a sequence of papers, have material and refs more directly related to your interests. | |
| Mar 11, 2022 at 22:22 | comment | added | Gottfried Helms | (...) The visual example with the 31-periodic point is on page 8. | |
| Mar 11, 2022 at 22:11 | comment | added | Gottfried Helms | (...) It is simple to locate n-periodic points near the fixpoints with $n=64$,$n=256$ ... or what you want. The disk around the fixpoint (let's use the primary with positive imaginary value) which has no n-periodic points seems of epsilon-size only. Or am I perhaps basically wrong with my opening doubt in the first part of this comment? | |
| Mar 11, 2022 at 22:07 | comment | added | Gottfried Helms | TomCopeland - I'm not firm in complex analysis; but concerning the "for s and t small enough" again, I think I've it correct, that the range of analycity of the function in the neighbourhood of a fixpoint cannot be larger than the distance to the next fix/n-periodic point. I've shown in my essay on my findings about n-periodic points that it is possible to locate n-periodic points arbitrarily near to the fixpoints, for instance the example of a 31-periodic point in go.helms-net.de/math/tetdocs/periodic_points_compact.pdf ... | |
| Mar 11, 2022 at 15:01 | comment | added | Tom Copeland | See eqn, 2.3.19 on p. 67 of K, C, & G for the associated Abel equation for the special linear fractional transformation and Thm. 8.5.3 on p. 347 for a relation to the Julia eqn. | |
| Mar 11, 2022 at 14:12 | comment | added | Tom Copeland | See also p. 46 and 47 of Iterative Functional Equations by Kuczma, Choczewki. and Ger. | |
| Mar 11, 2022 at 8:02 | comment | added | Tom Copeland | @GottfriedHelms, I included some basic examples in more detail to illustrate the analysis. | |
| Mar 11, 2022 at 8:01 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Examples in more detail of basic analysis
|
| Mar 9, 2022 at 17:07 | comment | added | Gottfried Helms | TomCopeland - thank you very much for your detailed answer! Unfortunately it contains too much material which I am still unfamiliar with, and I don't know, whether I'll ever get the spirit again to involve appropriately. If I can and find some way through I'll come back to this another day - this comment is just to tell you today I'm yet thankful for your kind workout w.r. to my specific question. | |
| Mar 7, 2022 at 21:19 | comment | added | Tom Copeland | @GottfriedHelms, hope I've addressed you comments correctly in the edit as to my meaning of small for the domains of $s$ and $t$ although, as my example indicates, the domains could extend far beyond that allowed in the differential-shift characterization. | |
| Mar 7, 2022 at 20:41 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Example added
|
| Mar 7, 2022 at 19:44 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Addressed comment
|
| Mar 7, 2022 at 10:00 | comment | added | Gottfried Helms | Hmm, concerning the formula $FL(FL(z,s),t) = FL(z,s+t)$: in my question mathoverflow.net/q/391772/7710 I've discussed the observation on periodic points, by which $FL(FL(z,s),t) \ne FL(FL(z,t),s) $ and thus $FL(FL(z,s),t) \ne FL(z,s+t)$ if $z$ is a n-periodic point and $s=n$ is the period-length. For me this problem/special case is still not clear/not resolved with the given answers, and it would be instructive, if some required restrictions on "for $s$ and $t$ small enough" would be expressed more explicite. (Hope I'm not asking for complete trivia and/or nonsense here ...) | |
| Mar 7, 2022 at 0:42 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Corrected a typo in a formula
|
| Mar 7, 2022 at 0:27 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Related answer to Newton interpolation
|
| Mar 6, 2022 at 16:35 | history | edited | Tom Copeland | CC BY-SA 4.0 |
corrected title of a book
|
| Mar 6, 2022 at 8:02 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Noted Schroeder's second paper and two Wiki links
|
| Mar 6, 2022 at 0:00 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Supporting refs
|
| Mar 5, 2022 at 21:39 | history | answered | Tom Copeland | CC BY-SA 4.0 |