Talk:Sharkovskii's theorem

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This article is within the field of Dynamical systems.


Can someone point me to an accessible version of the proof of this theorem ? I have tried looking on arxiv etc. to add it to the page, but have been unlucky so far. AmarChandra 17:00 May 23, 2004 (UTC)

stable cycles appear in Sarkovskii order in the bifurcation diagram, starting with 1 and ending with 3, as the parameter is varied. - I doubt that this is true. Isn't there bifurcation after 3, leading to stable cycles of period 6 for slightly larger parameters?

Also, Sarkovskii's theorem doesn't say anything about the way the period of stable cycles change as the parameter changes, so I think this statement, even if true, confuses the matter. AxelBoldt 01:46 Sep 30, 2002 (UTC)

There are stable cycles of various orders, including 5 as well as 6, 12, ..., after the 3-cycle. What counts is the first time the cycle appears. -phma

Oh, ok, I'll put the statement back then. AxelBoldt 21:26 Dec 22, 2002 (UTC)


We should move this term Sarkovskii's theorem to proper term Sharkovsky's theorem, since it is named after Ukrainian mathematician Oleksandr Mikolaiovich Sharkovsky and hence goes the English spelling. Best regards. --XJam 12:27 Dec 17, 2002 (UTC)

Google gives about three times as many hits for Sarkovskii's theorem than for Sharkovsky's theorem, so I suggest we leave the article at the more common spelling. AxelBoldt 21:26 Dec 22, 2002 (UTC)

I disagree in full. Bad habit. Don't mind the Google. Thousand times spoken lie becomes a truth. If Google is wrong, why should be Wikipedia then too. And still, if we translate his first name in English, we should write Olexandr and, I guess, not Oleksandr. But as it seems, nobody cares that. Nevertheless we should be even more precise here. That is my strong opinion. --XJam 23:32 Dec 23, 2002 (UTC)
Why do you think that Sarkovskii is wrong and Sharkovsky is right? AxelBoldt 02:49 Dec 24, 2002 (UTC)
I believe the spelling Sarkovskii comes from other languages than English, probably mostly from French and German language. Have you checked searching Sharkovsky just in Google's English pages? I haven't. But I guess it would give more terms than Sarkovskii. In Slavic languages a letter "s" is completely different from a letter "sh". I tried to find a person, who is responsible for this theorem under Sarkovskii, but I failed. It was just my lucky guess that I really found him. But I might be wrong after all too. I am just trying to be accurate as posible as I can. Someone is more careful regarding strictly math terms and someone regarding related math terms as names, surnames, birthplaces and such are. --XJamRastafire 10:36 Dec 24, 2002 (UTC)
Actually, I've seen Šharkovksii for the Ukrainian spelling. I agree that the best English spelling is probably Sharkovsky. Also, I have seen some experienced Wikipedians argue that articles about theorems should be named X theorem rather than X's theorem, although I am not sure I agree. One thing seems clear: whatever the name of the article, someone should ensure that there are suitable redirects from the other candidates. ---CH 21:02, 13 May 2006 (UTC)

On both his personal web page and his page at the Ukrainian Academic of Sciences he spells it Sharkovsky. A survey of my collection of textbooks on dynamical systems---all in English---reveals that Charkovsky, Sharkovskii, Sharkovsky, and Sarkovskii are all in use. But since Sharkovsky spells it Sharkovsky, it seems like Sharkovsky is the way to go. Dave Feldman (talk) 00:00, 14 July 2011 (UTC)

Clarification needs clarity[edit]

The article says:

"3, 5, 7, 9, ... ,2·3, 2·5, 2·7, ... , 2^2·3, 2^2·5, ..... , 2^4, 2^3, 2^2, 2, 1.

We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, etc., . . ."

The problem with this "clarification" is that many people unfamiliar with Sharkovskii's Theorem will wonder if this "etc." continues with 6 times the odds or with 8 times the odds -- something the "clarification" doesn't clarify.Daqu (talk) 03:00, 18 March 2008 (UTC)

Article name consensus?[edit]

Was there ever consensus on the title of this article? Personally, I'd prefer Sarkovskii, but I'm not going to get into that. The preferred name is debatable, but that the spelling should be consistent throughout the article is not, so I've changed all "Sarkovskii"s to "Sharkovskii"s since that is the current page title. Tcnuk (talk) 13:11, 2 March 2010 (UTC)

Burns/Hasselblatt paper[edit]

I added a link to a draft paper by Burns and Hasselblatt about the theorem. I put it in as an extlink rather than a reference since I don't know if it has been formally published. An earlier version[1] was submitted in 2007. Both links are from Richard Lipton's recent blog post[2] relating the theorem (and some other topics) to computational complexity. (talk) 22:40, 27 February 2011 (UTC)

Well-ordering - relevant?[edit]

The recently-added sentence: Note that this ordering is not a well-ordering, since the set \{2^k \ \mid\ k \in \mathbb{N}\} doesn't have a least element seems to be true, but I can't see any relevance the article - should it be deleted? --catslash (talk) 23:45, 30 December 2012 (UTC) I think in a well-order you can 'only go to infinity once', so there wouldn't be much significance to the fact a function has only finitely many periodic orbits. Otherwise, all functions with infinitely many periods would have every period. Gred Sixteen (talk) 13:14, 25 August 2014 (UTC) Pretty sure that \{2^k \ \mid\ k \in \mathbb{N}\} has a least element.