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 irrational iterate of 2z(1-z) BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 08/13/2010, 06:47 AM 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 Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 08/14/2010, 03:58 AM (08/13/2010, 06:47 AM)BenStandeven Wrote: 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 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/09/2014, 10:16 PM 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. --- 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. --- 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 « Next Oldest | Next Newest »

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