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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. « Next Oldest | Next Newest »

 Messages In This Thread irrational iterate of 2z(1-z) - by BenStandeven - 08/13/2010, 06:47 AM RE: irrational iterate of 2z(1-z) - by BenStandeven - 08/14/2010, 03:58 AM RE: irrational iterate of 2z(1-z) - by tommy1729 - 08/09/2014, 10:16 PM

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