(06/10/2011, 11:00 PM)sheldonison Wrote: Nicely stated. I don't think we know what happens on the Shell Thron region itself, except at base eta. For bases<>eta, on the boundary, we have a repelling fixed point, and a parabolic fixed point.
- Sheldon
This is a repost. The question is, at the Shel-Thron Boundary, can we develop a merged Fatou like mapping from the two fixed points? Originally, I was going to post that there are two fixed points on the Shel Thron boundary, but that only one of these could be used to develop an analytic superfunction; except now I'm pretty sure that both can be used to develop an analytic superfunction. Imagine following a circular arc around \( \eta \) starting from base \( \sqrt{2} \) and continuing along to base \( 2\eta-\sqrt{2} \). Then along this arc, the repelling fixed point which starts at L2=4 morphs seamlessly, and remains a repelling fixed point for the entire path, and becomes the complex conjugate fixed point, with Im(L2)<0. The superfunction from L2 can be calculated anywhere on this path, with known methods. The attracting fixed point starts out at L2=2. But when you cross the Shell Thron region (at the base given below), the attracting fixed point becomes neither attracting, nor repelling, nor parabolic. It becomes a neutral circular fixed point, with a real Period. Because the absolute value of the log of L1=1, this fixed point is not repelling, and not attracting, but rather circular.
From the wiki page, "The boundary of the Shell-Thron region ... where |log(L1)|=1".
Code:
Results at the Shel Thron boundary
base = 1.4749205843145635852862061112 + 0.0034981478091858060388588917267i
L1 = (circular) 2.4372341100745143830388282062 + 0.82552153847929520132720783617i
L2 = (repelling) 2.5206834347439820199663539286 - 0.87677210299493726158947155475i
|log(L1)|=1.0000000000000000000000000000 (circular)
|log(L2)|=1.0371394570002141836820569537 (repelling)
Period1= 18.886173137606161572331556476 (real period)
k = 0.33268705424861870835170403130i
Period2= -18.886314043722660117719560870 + 2.0956661382058037697500756852i
Now, assume that f is the superfunction from the L1 circular fixed point. If you take a point very close to the L1 fixed point, and use that sequence of points, all very close to the L1 fixed point, to try to develop a Superfunction, then you will find that the sequence of points has a real period, which I originally thought was nonsense, but now I think it is probably possible to develop an analytic superfunction at the L1 fixed point, although I don't know how to calculate it yet.
\( f(z)=L1+\delta \)
\( f(z+1)=B^{f(z)}=B^{L1+\delta} \)
\( B^{L1+\delta}\approx L1+\delta\log(L1) = L1+\delta\exp(\log(\log(L1))) = L1+\delta \exp(k) \) where \( k=\log(\log(L1)) \), where \( \Re(k)=0 \) (as shown below)
\( f(z+2) \approx L1+\delta\exp(2k) \)
\( f(z+3) \approx L1+\delta\exp(3k) \)....
It turns out that k is an imaginary number, \( k=\log(\log(L1)) \), where by definition \( |\log(L1)| \)=1. K is linked to the Period by the equation
\( k=\log(\log(L1))=2\pi i/\text{Period1} \).
Note that \( \Re(k)=\Re(\log(\log(L1)))=0 \), since |log(L1)|=1. So then k is an imaginary number, and Period1 is a real number.
\( f(z) \approx L1+\exp(zk) \), where exp(zk) decays to zero as \( \Im(z) \) increases. My conjecture is that this might work with an infinite sequence of terms, all decaying as \( \Im(z) \) increases.
\( \text{SuperFunction}_{L1}(z) = \sum_{n=1}^{\infty} a_n\exp(2n\pi iz/\text{Period1}) \)
So then we would have
\( f(z+\text{Period1}) \approx f(z+\exp(2n\pi i z/ \text{Period1})) \approx f(z+\exp(2\pi i)) \approx f(z) \), which is an exact real periodic equation, if an infinite number of terms is used for the superfunction. When I used only one term for the real periodic superfunction, the error term had twice the frequency, and was magnitude squared (smaller, as imag(z) increased). So, the question is, how to calculate the infinite sequence of terms, and does this work? At the Shel-Thron boundary, the other fixed point has \( \Im(L2)<0 \). Is it possible to somehow merge the two fixed point functions with a Fatou merger, or with a Kneser mapping?
- Sheldon
