[seqfan] Re: Sequence related to jittery function

[David Seal]

Using (b-a, 2n-b-a) rather than (b-a, 2n+1-b-a) for the case that b+a > n would mean that the image of the map would:

* only include (a,b) pairs for which a and b have the same parity, and thus not include all elements of X_n (other than in the degenerate case n=1);

* include (a,b) pairs that are outside X_n because they would have two equal components, since the image of (a,n) would be (n-a,n-a).

So while adding the extra 1 looks a bit artificial, it or something similarly artificial-looking is needed to make the map a permutation.

Best regards,

David


> On 11/02/2021 18:00 Andrew Hone <a.n.w.hone at kent.ac.uk> wrote:
> 
>  
> Dear Jens,
> 
> It may not help much, but you could try to consider each map j_n in a slightly wider context, as a piecewise affine map of the plane R^2. In other words, the definition of j_n makes sense for any pair of real numbers (a,b), and it is just a linear map for a+b<=n, and a composition of a linear map and a translation for a+b>n.
> 
> You say that the map is "not too artificial". The thing that looks strange to me is the fact that you add 1 to the second component: it would be more natural to take
> 
> (b-a,2n-b-a)
> 
> as the formula for a+b>n, because then the two formulae would agree when a+b=n, and then the map would be continuous on R^2, and might behave better. It might be worth comparing the integer orbits of your map with this one to see if there is anything special about the distribution of periods.
> 
> Another map to compare with is the linear map on the torus:
> 
> (a,b) |-> (b-a,b+a) mod n.
> 
> There is quite a lot of literature in dynamical systems/number theory about the latter.
> 
> All the best,
> Andy
> ________________________________
> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Jens Voß <jens at voss-ahrensburg.de>
> Sent: 11 February 2021 17:20
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Sequence related to jittery function
> 
> 
> Let n be an integer greater than or equal to 2, and define
> 
>    X_n := { (a, b) | 1 <= a < b <= n }
> 
> to be the set of ordered pairs of distinct positive integers <= n.
> 
> For every such pair (a, b) € X_n, let
> 
>                   /  (b-a, b+a)      if  b+a <= n
>    j_n ((a,b)) := |
>                   \  (b-a, 2n+1-b-a) otherwise
> 
> It is easy to see that
> 
> (I)  j_n ((a,b)) € X_n for all (a, b) € X_n, i.e. j_n: X_n -> X_n and
> (II) j_n is injective.
> 
> Since X_n is finite, j_n is bijective, i.e. a permutation of X_n.
> 
> I am interested in the cycle representation of j_n; in particular,
> for every positive integer k, I would like to know the set of values
> of n for which j_n contains a cycle of length k.
> 
> It is not too hard to show that
> 
> (1) j_n has a cycle of length 1 (i.e. a fixed point) iff n == 2 (mod 5).
> (2) No n exists for which j_n has a cycle of length 2.
> (3) j_n has a cycle of length 3 iff n == 3 (mod 7) or n == 6 (mod 13).
> 
> I suppose that larger cycle lengths can be classified in a similar way,
> but calculations quickly become rather tedious, so I used a simple
> computer program to calculate the cycle lengths of j_n for all values
> of n up to 3000 (after that, the program gets really slow).
> 
> It appears that most smaller cycle lengths (except of course for 2) do
> come up eventually, the first stubborn exception being the number 27.
> However, the fact that this number (as well as the subsequent exceptions
> 38, 41, 53, 57, 62, 98, 103, 122, ...) does not show up as a cycle
> length of an j_n with n <= 3000 does not necessarily mean it can't do so
> for some larger n.
> 
> So the questions I am facing right now are: Is 27 a cycle length of an
> f_n for some n? If so, what is the smallest such n? How about the other
> exceptions I mentioned? How many numbers k > 2 do not occur as cycle
> lengths; are there infinitely or finitely many or none at all?
> 
> Does anyone have an idea on how to tackle these kind of problems? Has the
> function j_n perhaps already been studied by someone - it does not look
> too artificial to me!
> Ideally, if every integer > 2 is in fact a cycle length of some j_n, I'd
> like to submit the sequence of the smallest such n - currently, I am
> limited to the values for 3 through 26 and don't even know whether it
> can be extended beyond that point or whether it has a "hole" at k=27.
> 
> Thanks and best regards
> Jens
> 
> 
> 
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