irrational iterate of 2z(1-z)

[tommy1729]

First Tommy-Ben Conjecture :

let a,b,c be positive integers.

Let X,Y be positive irrational numbers that are linearly independant but they are NOT algebraicly independant.

Nomatter what X,Y are , there is NO non-Möbius closed form function f(x) such that

f^[a + b X + c Y](x)

is also a closed form for every a,b,c.

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Ben's OP was an example where f^[a + b X](x) had a closed form for every a,b. ( X was lb(3) )

Im very very convinced of this conjecture.

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Second Tommy-Ben conjecture :

Let a_i be positive integers and X_i be linear independant positive irrational numbers.
Let n be an integer > 0.

If f^[a + a_1 X + a_2 X_2 + ... + a_n X_n](x) is a closed form for every a_i then the superfunction of f is a composition of at least 2 functions with an addition rule and the X_i are all of the form log(A_i)/log© where the A_i are integers and C is a constant.


regards

tommy1729