[seqfan] Re: Sequence related to jittery function

[Andrew Hone]

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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