irrational iterate of 2z(1-z)

[BenStandeven]

From examining the superfunction for 4z(1-z) about z=0, \( sin^2(2^n) \) (one of the few elementary examples for a non-Mobius function), we get:
\( sin^2(2^{n + \log_2 3}) = sin^2(3 \cdot 2^n) = (3 cos^2(2^n) sin(2^n) - sin^3(2^n))^2 = (3 sin(2^n) - 4 sin^3(2^n))^2 = sin^2(2^n) (3 - 4 sin^2(2^n))^2 \)

So [one of] the \( \log_2 3 \) iterate of \( 4z(1-z) \) is \( z(3 - 4 z)^2 \). Putting these functions into the "Mandlebrot" form by conjugating with \( z = -w/4 + 1/2 \), we get that the \( \log_2 3 \) iterate of \( w^2 - 2 \) is \( -3w + w^3 \).

I was hoping to include some pictures of the Julia sets of these two functions, but I don't have ready access to a program that can draw general cubic sets (such as the old Autodesk Chaos program). So I'll try again in the morning, using Fractint.

[BenStandeven]

I was hoping to include some pictures of the Julia sets of these two functions, but I don't have ready access to a program that can draw general cubic sets (such as the old Autodesk Chaos program). So I'll try again in the morning, using Fractint.

Here is the Julia set of \( z^2 - 2 \):    

And here is the set for \( z^3 - 3z \):    

They don't seem to be displaying on my machine, though.

[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