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Natural cyclic superfunction are superfunction that naturally extend from integer iterates to real ones.
Example
F(2x) = 2 F(x) sqrt( 1 - F(x)^2 ) = g(F(x)).
F(1) = sin(1) makes the solution natural : sin(x).
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Tommy1729
Posts: 1,640
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Funny thing is this is Well studied for the complex numbers.
We can probably learn from considering fractals and chaos on the complex plane and then take the real part ...
Mick has posted this on MSE and MO btw.
Maybe that Will help.
I lack time but intend to work and answer this !
Optimistic.
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Tommy1729
Posts: 1,640
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Joined: Feb 2009
See
http://math.stackexchange.com/questions/...nal-period
Where our friend mick has posted the question on MSE.
Thanks mick.
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Tommy1729
Posts: 1,640
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Joined: Feb 2009
In the first post I starters with a variant of the logistic map.
Guess that is obvious.
I started considering
F(2x) = g(F(x)).
Mick started with F(x+1) = g(F(x)).
2 forms of the same idea.
Just like fake function theory we can see this as an intresting regression problem.
Mick's latest post captures the idea very well !
http://math.stackexchange.com/questions/...-a-n-a-n-1
Although a specific question, im confident an answer will provide the necc insights for many cases.
Regards
Tommy1729