Talk:Sharkovskii's theorem

Proof

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)

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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)

Spelling

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)

I was just thinking about this. While I was at university, I was taught that the name is known by different spellings, similarly to Chebyshev, apparently as a result of different transliterations from the Cyrillic.

IIRC, the worksheet on which I was taught about his theorem had it as Šarkovskii (except where explicitly listing the different spellings). But I don't remember now what other spellings it listed. On the English WP I see Sharkovsky and Sharkovskii. The only other form I see in the sources is Sarkovskii. But I see a few others in other language Wikipedias – Charkovski, Scharkowskyj, Szarkowski, Szarkowskiego. That said, the last of these appears to be a possessive or genitive form. If I put the Cyrillic into Google Translate, it's a bit weird: Олекса́ндр Миколайович Шарко́вський becomes Oleksandr Mykolayovych Sharkovskyi, but Шарко́вський by itself becomes Sharkovsky.

Are you sure about Šharkovksii? This seems erroneous - I would expect 'Š' to already make a /ʃ/ sound, rendering the 'h' spurious.

It would be good if we could find something about the different spellings to write on WP, probably on the Oleksandr Mykolayovych Sharkovsky page rather than here. — Smjg (talk) 01:19, 12 September 2022 (UTC)

Clarification needs clarity

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?

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)

As the mathematician's name is Александр Николаевич Шарковский, I see little room to prefer the spelling Sarkovskii.--Hagman (talk) 14:45, 26 November 2016 (UTC)

Burns/Hasselblatt paper

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. 71.141.88.54 (talk) 22:40, 27 February 2011 (UTC)

Well-ordering - relevant?

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. — Preceding unsigned comment added by 139.67.20.29 (talk) 9 September 2014 (UTC)

$\{2^{k}\ \mid \ k\in \mathbb {N} \}$ has a least element under the usual natural number order, but under the Sharkovskii order it doesn't, since Sharkovskii orders ... < 8 < 4 < 2 < 1. Which is the content of the wiki remark I guess. Jan Burse (talk) 21:20, 20 November 2016 (UTC)

The misleading note has been removed by Loraof on 12 September 2015 in Special:Diff/680708792, then the topic was expanded and explained by CiaPan on 13 February 2019 in Special:Diff/883119856. --CiaPan (talk) 09:18, 24 July 2019 (UTC)

Bad counterexample

The function $f(x)=(1-x)^{-1}$ is named as a counterexample showing that continuity is needed. However, to me it seems more importantly a counterexample that shows that the $I$ in $f\colon I\to I$ must be an interval, i.e., a connected subset of the reals. After all, $f(x)=(1-x)^{-1}$ is a continuous map $f\colon I\to I$ if one allows $I=\mathbb {R} \setminus \{0,1\}$ (as well as if one allows $I=\mathbb {R} \cup \{\infty \}$). That dropping continuity allows arbitrary behaviour is clear anyway.--Hagman (talk) 14:39, 26 November 2016 (UTC)

@Hagman: I have fixed and extended the counter-example in Special:Diff/882409296. Hope it is both redable and correct now. :) --CiaPan (talk) 13:38, 10 February 2019 (UTC)

Sequence

Is Sharkovski order an integer sequence ? --Adam majewski (talk) 13:48, 11 November 2021 (UTC)

@Adam majewski: Nope. It's not an integer sequence because it is not even a sequence. --CiaPan (talk) 20:51, 11 November 2021 (UTC)

@CiaPan:A GRAPH THEORETIC PROOF OF SHARKOVSKY'S THEOREM ON THE PERIODIC POINTS OF CONTINUOUS FUNCTIONS by CHUNG-WU HO AND CHARLES MORRIS : "Sharkovsky's theorem says that a function f:R—>R having a point of period m, must also have points of period n precisely when m precedes n in the above sequence." So it calls it a sequence = an an ordered list of integers. --Adam majewski (talk) 15:19, 14 November 2021 (UTC)

@Adam majewski: That's wrong. A sequence is commonly defined (→Sequence#Definition) as a function on all natural numbers (infinite sequences) or some initial segment $1,2,\ldots n$ of them (finite sequences). Sharkovskii's theorem defines an order in a set of all natural numbers, which is infinite. An infinite sequence has no last element, because natural numbers (a set of indices) have no last element. However, the ordering has a last element, namely 1. As such it's not a sequence. Q.E.D. --CiaPan (talk) 18:27, 14 November 2021 (UTC)
Edited. --CiaPan (talk) 21:20, 14 November 2021 (UTC)

Adam majewski, here is another approach. Consider the first row of the Sharkovskii's ordering description. It enumerates all odd natural numbers (except 1):

$3\quad 5\quad 7\quad 9\quad 11\ \ldots \ (2n+1)\cdot 2^{0}\ \ldots$

Odd numbers make an infinite sequence, thus exhausting all natural numbers as a set of indices. Anyway, then another infinite sequence goes, appended after the first one – those are all $(2n+1)\cdot 2.$ And then, all $(2n+1)\cdot 2^{2}.$
And an infinite sequence of infinite sequences of even numbers follows. Plus the terminating $1$ at the end (!).

The above construction makes it impossible to define a position (index) of any even term, as there is an infinite set of numbers preceding it (all odd numbers, at least). This contradicts a definition of a sequence as a function, that is an assignment of values (terms) to arguments (indices from a set of natural numbers). --CiaPan (talk) 10:02, 7 December 2021 (UTC)

@CiaPan: Right. Thx. --Adam majewski (talk) 16:09, 8 December 2021 (UTC)

References