Posts: 1,491

Threads: 355

Joined: Feb 2009

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).

Regards

Tommy1729

Posts: 1,491

Threads: 355

Joined: Feb 2009

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.

Regards

Tommy1729

Posts: 1,491

Threads: 355

Joined: Feb 2009

See

http://math.stackexchange.com/questions/...nal-period
Where our friend mick has posted the question on MSE.

Thanks mick.

Regards

Tommy1729

Posts: 1,491

Threads: 355

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