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Do functions which, when composed with themselves, are equivalent to the identity function (i.e. functions for which $f(f(x)) = x$ in general) have a name and if so, what is it?

Additionally, am I correct in saying that a such function has a splinter of two, or is it perhaps splinter of size 2 or something else entirely? Or could I say that a such function has an orbit of size 2?

rschwieb
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jaymmer - Reinstate Monica
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1 Answers1

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These are involutions. The orbits of an involution all have size $1$ or $2$.

What is a splinter?

Did
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  • I read "splinter" on the wikipedia article on iterated functions, but I don't think I even read the sentence correctly. – jaymmer - Reinstate Monica Feb 08 '14 at 11:16
  • "Splinter" seems to go back to a suggestion by Ullian, J. in Splinters of Recursive Functions, The Journal of Symbolic Logic, Vol. 25, N. 1, March, 1960, pp. 33 - 38. I never met the term before and it does not seem to me to be much in use in any community (but I might be wrong). – Did Feb 08 '14 at 11:24
  • I have some old lecture notes which call the splinter the sequence of functions you get from repeatedly applying the function, i.e., ${f(x), f(f(x)), f(f(f(x))), \ldots}$, so I suppose a “splinter of size 2” might mean this sequence only has two distinct elements, or something similar. – alexwlchan Feb 08 '14 at 12:03
  • I'd like to up vote, but that second sentence is blocking me. I've never seen the word orbit used to describe the set if powers of something. It's certainly not the standard use in group theory. – rschwieb Feb 08 '14 at 12:06
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    @alexwlchan Sequence of values, not sequence of functions. – Did Feb 08 '14 at 12:08
  • @rschwieb Please stay blocked. Orbits in groups: http://en.wikipedia.org/wiki/Group_action#Orbits_and_stabilizers – Did Feb 08 '14 at 12:12
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    Dear @Did: Oh! I gather now you are thinking of the action of the map on its set, rather than some orbit of the involution in the group of transformations. I've got a tip: given the choice between a) explaining your thinking and b) returning a sarcastic remark with a link to basic algebra, choosing a) will be both more polite, faster, and more helpful to your peers (and students, if you have them.) Regards. – rschwieb Feb 08 '14 at 14:02
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    @rschwieb Thanks for the tip. To summarize, you asserted something false that anybody can disprove in one minute using google and wikipedia, and when I provide the link, you lecture me on manners. Well... By the way, who was sarcastic first? – Did Feb 08 '14 at 18:07