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.
\( sin^2(2^{n + \log_2 3}) = sin^2(3 \cdot 2^n) = (3 cos^2(2^n) sin(2^n) - sin^3(2^n))^2 = (3 sin(2^n) - 4 sin^3(2^n))^2 = sin^2(2^n) (3 - 4 sin^2(2^n))^2 \)
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.