07/05/2011, 01:09 AM
(This post was last modified: 07/05/2011, 01:53 AM by sheldonison.)
(07/04/2011, 10:56 PM)tommy1729 Wrote:(07/02/2011, 10:32 PM)tommy1729 Wrote: in particular i still dont know why 1.729 i ^1.729 i ^ ... 0.5 + 0.5 i gives a circle and that bothers me.
nobody is telling me ...
I can't follow your latest tommysexp equations right now, but the part about 1.729i is an approximation for b=1.712936...i, which is the base on the Shell Thron boundary region with real(b)=0. All bases on the Shell Thron boundary with an irrational period are going to give a circle (as long as the irrational period is a Brjuno number). The definition is the abs(log(fixed_point))=1, which leads to a neutral fixed point. Look up Siegel Disc on Wikipedia, for some background. Also, recent discussion has some really good papers referenced in it (on the Brjuno number). Each iteration circles around the h(z) function (inverse Schroder function), so how circular the path is depends on the magnitude of the initial point (0.5+0.5i)-L, which for this case is L~= 0.390842217 + 0.4586753i, so 0.5+0.5i is pretty close to the fixed point, so the path is fairly circular.
1.729*I is a very very slightly repelling fixed point, that will eventually diverge, abs(log(L))=1.00244, but it would take a long time.
- Sheldon