Talk:Iterated function

Redirected page
Function Iteration was redirected to Function composition. Should this article redirect there as well (after any necessary merging)? - dcljr (talk) 18:58, 11 November 2005 (UTC)


 * No. Please note this comment should have been placed at the bottom of the talk page, in the first place. Cuzkatzimhut (talk) 18:57, 28 December 2015 (UTC)

Merge with Recurrence relation?
It appears that the articles Recurrence relation and Iterated function are about the same thing. Is there any objection to merging them? Which should be merged into which? Duoduoduo (talk) 22:52, 28 May 2010 (UTC)

See discussion at talk:Recurrence relation. —Preceding unsigned comment added by Jowa fan (talk • contribs) 06:19, 1 June 2010 (UTC)

Someone can help?
Surely the expressions can be by far non rigurous, but can be someone to look my notes on and improve that or comment about? — Preceding unsigned comment added by 80.25.164.213 (talk) 20:06, 20 July 2012 (UTC)
 * I sense you are in the wrong article, except perhaps for the brief summary remarks of the "Conjugacy" section. You are replicating, in somewhat idiosyncratic language, the continuous iteration orbit theory of Schröder's equation. Possibly  is useful. Cuzkatzimhut (talk) 20:16, 20 July 2012 (UTC)

My little contribution
From my time as student I develop some aproximation to this field. Maybe some ideas can be useful. — Preceding unsigned comment added by 80.25.164.213 (talk) 10:27, 20 August 2012 (UTC)
 * Your compositional index eqn is Abel's equation, with a standardized theory: You are constructing Koenigs function for Schröder's equation.Cuzkatzimhut (talk) 10:56, 20 August 2012 (UTC)

Existence and uniqueness of fractional and continuous iterates
I think the article should say a bit more about the question of existence and uniqueness (or lack thereof) of fractional and continuous iterates. At the moment it simply says "In some instances, fractional iteration of a function can be defined", which is not tremendously illuminating. 86.160.216.252 (talk) 13:31, 24 October 2012 (UTC)


 * My own sense is that this article here is a popular entryway into the subject, and anyone more seriously interested and more mathematically inclined to worry about existence and uniqueness would have moved on to Schröder's equation and thence Koenigs function, where such issues belong, a while ago &mdash; if not Böttcher's equation and Abel's equation. I assume the reader of this article is an engineer or an undergraduate, the reader of Schröder's equation a graduate student, and that of  Koenigs function,  a mathematically sophisticated reader.  If one wished to adduce the first Kuczma book reference after "defined", that might useful. But adducing the Szekeres, etc... references that address the problem properly and completely would be overkill, and could only alienate the "first contact" reader seeking something compact and practical....Cuzkatzimhut (talk) 15:03, 24 October 2012 (UTC)
 * I agree that this article should not go into enormous technical detail about this topic. I just think it should say a bit more than it presently does about the two questions "Do fractional/continous iterations always exist (e.g. for "sensible" continuous functions)?" and "If they exist, are they unique?". It only needs to be a handful of lines, with perhaps a couple of examples. I think this is of general interest to the curious reader. 86.160.216.252 (talk) 17:15, 24 October 2012 (UTC)
 * Indeed, a few evocative illustrations might be helpful, if simple. Cuzkatzimhut (talk) 18:27, 24 October 2012 (UTC)

fk notation
There was a section in inverse function related to $f^{ 2}$, $f^{ k}$, and so, that I trimmed due to obvious WP:stay on topic concerns. Can somebody reuse this stuff here? Incnis Mrsi (talk) 15:35, 3 July 2013 (UTC)

Verifying some formulas for fractional iteration
The section Some formulas for fractional iteration is very similar to my own unpublished results. My question is, can anyone verify the formulas are in a peer reviewed published work? Daniel Geisler (talk) 20:21, 13 May 2014 (UTC)
 * My notes too, of course. Please sign your posts. You are referring to the off-the-bat power series expansion around a fixed point that, I trust, you saw in the History of the article, User:Drschawrz adduced in August 2011? This is the frontal--and inefficient--assault to the problem, that, mercifully, Schroeder has provided the more systematic solution to a century and a half ago. Try reproducing his results on the table of section 9 (Examples) that way! Conjugacy is of course the way to go.  Most standard courses on iterated functions and textbooks have, naturally, one version of them or another. You are unhappy with the Carleson and Gamelin 1993 text? Cuzkatzimhut (talk) 00:22, 14 May 2014 (UTC)
 * The issue is whether the Taylors series can be validated by a peer reviewed article in a published journal. Am I unhappy with the Carleson and Gamelin 1993 text? Actually I contacted one of the authors and he said he no longer worked with complex dynamics. My concern is that while the Classification of Fixed Points documents the Schroeder and Abel equations, it does it in a round about way and provides no explanation of why they are necessary. The power series expansion around a fixed point has interesting combinatorial properties besides providing a natural explanation for both the Schroeder and Abel equations. Daniel Geisler (talk) 06:54, 14 May 2014 (UTC)
 * You are evidently proposing rewrites of the Carleman matrix algorithm, which most of the references cited here hew to. An ambitious project. The Scroeder equation solves the project for somebody more advanced than the novice coming to this specific article here, as discussed above. Further technicalities would only obscure  the picture here---but you could test-drive your proposals on superfunction which needs your, and anyone's really!, help.  Are you sure you contacted the author User:Drschawrz  of these sections? Cuzkatzimhut (talk) 10:44, 14 May 2014 (UTC)

It's been a week and nobody has provided any reason to think that the section on fractional iteration can be validated through being published. Ironically I don't disagree with the results, I just think they should be published first. Daniel Geisler (talk) 11:22, 21 May 2014 (UTC)
 * ? What exactly is your point? You are proposing to delete section 6 until somebody finds a remote refereed attribution for the elementary examples? As I indicated, this would be detrimental to the article, but not a tragedy, as they are misleading in suggesting to the reader how professionals in the field actually solve such equations in practice. To me, the examples appear obvious--the first naive thing that crosses one's mind--albeit off the mark. Starting from Babbage and continuing with Schroeder, both in the 19th century, conjugacy is the answer. Check it out on the iteration orbit of, e.g. sin(x) as contrasted to the actual answer, e.g. in [the roots of sine]. However, at the end of the day, I see no actual harm in these examples. Cuzkatzimhut (talk) 12:59, 21 May 2014 (UTC)

Should this page be renamed to "Function iteration"?
I've rewritten the lead to make it make sense at least marginally. However, it struck me that the article seems to be more about the general notion of iterating functions (as a generaly activity, or subject of study) than about the specific subject of functions that are of the form f n. Therefore I think the name "Function iteration" would much better cover the contents than "Iterated function". Marc van Leeuwen (talk) 08:42, 18 February 2015 (UTC)


 * Of course your name is better, but handling dead links could be a nightmare. And links is what WP is all about. As a stopgap, you should make a redirect.
 * By the way, "splinter" is a standard term, used interchangeably with "Picard sequence" in less parochial corners of the broad and disparate user community, and is borrowed from recursive function theory, Ullian, J., Splinters of Recursive Functions, The Journal of Symbolic Logic, Vol. 25, N. 1, March, 1960, pp. 33 - 38.Cuzkatzimhut (talk) 11:32, 18 February 2015 (UTC)


 * A redirect Function iteration-->Iterated function is present since 2006. Perhaps the lead should be rephrased to cope with both titles. What about:
 * In mathematics, function iteration is the process of composing a function f: X → X (that is, a function from some set X to itself) with itself a certain number n of times. In this process, starting from some initial element of X, the output of f is fed again into f as input, and this process is repeated. The result of function composition is again a function fn: X → X, it is called an iterated function.
 * or something similar? The sentence "The process of repeatedly applying the same function is called iteration." could be moved down e.g. to the "Definition" section, and adapted to e.g. "Function iteration is a particular example of iteration in general." if it is considered worth to be kept at all. In contrast, the possibility of fractional iterates should be mentioned already in the lead, I think. - Jochen Burghardt (talk) 17:31, 18 February 2015 (UTC)

Slight Ambiguity
When the article says, "Note: these two special cases of ax2 + bx + c are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table." Does it mean that we only know of two closed form solutions, or has it been proven that there are no others? — Preceding unsigned comment added by 75.243.141.93 (talk) 21:20, 4 October 2015 (UTC)
 * Well, a "proof" for the absence of a closed form solution might be hard to conceive, mostly because "closed form" is a bit of a subjective label: it is predicated on agreement of what the "closed form" solutions of Schröder's equation may include. You may see, though, in conventional language, from the original Schroeder paper of 1870 that they really have to be just these two for the logistic map, and the Katsura and Fukuda paper pushes the envelope, but only comes up with trivial changes of variables. If you thought you found new ones which do not rely on implicitly defining new functions solving the Schroeder equation, you might well suggest them on this talk-page. Cuzkatzimhut (talk) 22:50, 4 October 2015 (UTC)
 * ik this is very late but what does the Katsura and Fukuda  say. 100.2.153.196 (talk) 02:00, 29 November 2023 (UTC)
 * also is there any way to access these papers without paying 100.2.153.196 (talk) 02:13, 29 November 2023 (UTC)

Complex Iteration
I've discovered that not only fractional iteration is possible, but also complex iteration. It's not as intuitive as fractional iteration, but it does make sense once you have a new model for what iteration means. Probably the simplest explanation is with orbitals in the complex plain: iterating to a real values generates the complex orbitals as paths. Iterating to imaginary powers, meanwhile, generates paths perpendicular to the orbitals at every point. You need the entire vector field of orbitals to know the new path. For example, iterating a linear function (multiplying by a constant) creates orbitals which are rays. Iterating to imaginary powers creates paths with are concentric circles. After that, complex iteration can be done by composing real and imaginary values. For a more precise derivation, I had to invent a new type of derivative, it looks at the instantaneous change in value at a point as the function iterates. The formula for this derivative is: f*(x) = lim n->0 (fn(x) - x)/n Which is equivalent to the derivative of f's Abel function. The key about this derivative is it has the property that fn* = f x n. This enables iteration to imaginary powers to be calculated directly. Ultimately I think it is equivalent to taking the analytic extension of the Abel function, but it's a bit more intuitive than Taylor series witchcraft. I've done the calculations to arrive at Euler's identity using this method rather than the typical one, and graphically it makes much more sense: once it's understood how complex iteration works, it's just geometry to find the imaginary value e must be raised to in order to complete the arc from 1 to -1. — Preceding unsigned comment added by 216.49.181.128 (talk) 15:34, 17 December 2015 (UTC)


 * Please heed & comply with above rubric, What Wikipedia is not, No original research.                   Cuzkatzimhut (talk) 19:17, 17 December 2015 (UTC)

Question:How did the editor get away with treating "n" like a regular power???
In one of the subsections, it says "When n is not an integer, make use of the power formula y n = exp(n ln(y))". This will not work at all and makes no sense, n is not a power, it is the number of iterations. That line should be deleted.

There's no reply button, so I'll just have to edit here to say that I don't see anything in the Curtright, T.L. Evolution surfaces and Schröder functional methods. That article does not address what I asked, and instead of using facts and evidence, someone trolled by editing my question to say the article did, so I am reporting them for a moderator to look into.

I have not seen anyone actually bring up any logical contention with this change, so in two days (from my time zone) I will delete that sentence. Posted by User:Leakdope without signature, 12/2018.

````Leakdope```` — Preceding unsigned comment added by Leakdope (talk • contribs) 04:09, 12 December 2018 (UTC)


 * Please sign your postings with four tildes, ~, as instructed by your edit session. Cuzkatzimhut (talk) 15:13, 11 December 2018 (UTC)

Placing the Iterated function and Tetration pages on more solid mathematical ground
Please see my comments at Talk:Tetration which is relevant to both articles. Daniel Geisler (talk) 18:53, 30 April 2019 (UTC)


 * Isn't this a bit scattershot? This article here is reasonably well sourced, and anything and all connections to tetration may reasonably be severed, probably usefully, no? Fractional iteration is a mature functional conjugacy field and amply sourced and documented. If you wished to delete the generic unnumbered "example" 6.2 sourced to the Tetration site, you should be welcome, after proposing specific defensible deletions: proceed! Cuzkatzimhut (talk) 19:34, 30 April 2019 (UTC)

Sorry, but I strongly disagree with the statement that fractional iteration is a mature functional conjugacy field. Papers are being published on extending tetration to the real and complex numbers solely on Abel's Functional Equation which is wrong because it is only valid for $$f(z)$$ once the coordinate system is shifted to a fixed point $$f(0)=0$$ then $$f'(0)=1$$ must be true. I am in the process of writing something up to send to these authors and the editors of the journals they published in.

If you review the older material from the beginnings of the Tetration Forum you will see them arguing that fixed points were not important, while my work starts with fixed points. So this work is inconsistent with Schroeder's Über iterirte Functionen, considered the first paper on dynamics.

Even though I try and track people and papers, both for myself and the community of mathematicians I communicate with. Maybe our different views of functional conjugacy are based on you being connected to people and papers I am unaware of. I'd appreciate any information you wish to share, either publically or privately. Thank you.

Daniel Geisler (talk) 20:45, 1 May 2019 (UTC)


 * Sorry this is not a forum. A combination of Schroeder's, Abel's, or Boettcher's equations, all linked to cover each other's blind spots have been used for over a century, and this is the somewhat dull attested work covered here. Again, this is not a forum, and not a Tetration venue. Cuzkatzimhut (talk) 21:23, 1 May 2019 (UTC)

I'm sorry then, I see that we will not be agreeing. This is not an appropriate place for me to benefit people. I'm fine with letting history judge the merits of our work here. So best wishes to the folks here.

Daniel Geisler (talk) 07:49, 2 May 2019 (UTC)

The article would be greatly improved
by the omission of the top illustration, which is overwhelmingly confusing.

It is confusing simply because it tries to convey about ten times as much information as one illustration can convey.

Often — and in this case — less is more.

If the illustration were replaced by a far simpler one, having almost no text, that would be a good thing. 2601:200:C000:1A0:5FB:D9A8:4BAF:D605 (talk) 19:34, 24 November 2022 (UTC)

Bad article
This article contains multiple passages with bad, confusing, and/or misleading writing. The top illustration is at least as confusing as any other bad illustration in Wikipedia, but even more so. Significant important relevant topics, like Koenigs's theorem as well as the central topic of embedding certain functions into a continuous flow, are left entirely unmentioned. And the passage about iterating f(x) =  (√2)x is a train wreck. - 2601:200:c082:2ea0:a874:6184:ede8:53c1 11:56, 21 May 2023‎


 * You are invited to flag each passage with an appropriate template, see Template:Inline cleanup tags for an overview, or even to improve the text yourself, cf. WP:Bold. Also, you may replace the top illustration by a better one, and suggest sections about unmentioned topics. - Jochen Burghardt (talk) 12:10, 21 May 2023 (UTC)

actual half iterate of x
if you use the f^0(x) axiom,as well as the exponentioation property of iterated functions you can conclude that the half itarate of x aka the identy map is itself. 100.2.153.196 (talk) 01:24, 20 November 2023 (UTC)


 * I guess there is an error in your proof. Can you give it in detail? - Jochen Burghardt (talk) 11:00, 20 November 2023 (UTC)
 * 1. g(x)=f^0(x)=x
 * 2.g^n(x)=nth iterate of f^0(x)=f^0n(x)=f^0(x)=x 168.100.171.2 (talk) 14:54, 20 November 2023 (UTC)
 * What are you going to prove, what is f, what is g? Why should 1. hold? Can you express 2. in a formal correct way, without English text appearing in expressions? - Jochen Burghardt (talk) 08:18, 21 November 2023 (UTC)
 * 1. holds as f^0(x) always equals x its a given in the deinition of an iterated function
 * 2. i had no notation that could do that but up, but i relized the wikiedia article has some
 * 3. revised proof
 * g(x)=x=f^0(x)
 * g^n(x)=(f^0)^n(x)=f^0(x)=x 100.2.153.196 (talk) 03:34, 22 November 2023 (UTC)
 * minor mistake instaid of jumping from f^0^n(x) to f^0(x) i should go to f^0n(x) 100.2.153.196 (talk) 03:35, 22 November 2023 (UTC)
 * Again: If anybody is supposed to follow your proof, you should bring it into a strictly mathematicel form, as found in typical math textbooks. You should initially state your claim. In the following proof, every step should be justified, as an assumption, from a definition, or as an inference of earlier results. The claimed property should be the last step. Like e.g. "I claim that for every ... we have ... . To prove it, let ... be ... . Then, we have ... = ... = ... by ... and ..., respectively. Hence, ... by ... . etc. ... Hence, we are done." - Jochen Burghardt (talk) 10:13, 22 November 2023 (UTC)