02/21/2013, 09:01 AM
(This post was last modified: 02/21/2013, 08:34 PM by sheldonison.)

(02/19/2013, 11:22 PM)bo198214 Wrote: Ya I figured out that the holomorphy mentioned in Shishikuras article is only for functions that have the fixed points rather vertically aligned.

I added the corresponding text from Shishikuras chapters here: http://math.eretrandre.org/hyperops_wiki...sition_A.1

Proposition A.1 is about this holomorphy. The whole proposition A.1 is applicable only for functions which are in the class , which corresponds to having the fixpoints rather vertically aligned.

In our case this probably means: as long as the base is outside the Shell-Thron-Region it depends holomorphically on the base.

But as soon as we pass the Shell-Thron-Boundary the both fixpoints collapse and hence the function is no more . And inside the STR I guess the fixpoints are rather horizontally aligned which also means its not in .

(Of course I always mean here the corresponding meaning of if the first fixpoint is not set to 0.)

Henryk,

So you think Shishikura's results only applies to points outside the main cardioid, where both fixed points are repelling???? For the Mandelbrot set, the "main cardiod" is exactly analogous to the Shell Thron boundary for complex exponentials. We did see some results for bases both inside and outside the Shell Thron boundary. On the boundary, one of the fixed points has a real valued period. But the two fixed points are still vertically aligned; I think.... Anyway, the results I saw seemed to analytically continue to results on the boundary itself (with a singularity at eta). But I was working only with the sexp(z) function, and I really didn't work with the slog, which would be analogous to Shishikura's perturbed fatou coordinates, but I don't understand his paper yet . I'm still fascinated by the possibility of solutions on the Shell Thron boundary itself, and in my posts last year, I did some calculations for a complex base with , which has a Period=5, and is a rationally indifferent function case on the Shell Thron boundary itself. But I think the two fixed points are still "vertically aligned", in that for the period=5 ShellThron boundary base, fixedpoint neutral=0.791130 + 1.10878i, and fixedpoint repelling= 0.422887 - 2.09203i, so perhaps Shishikura's results do apply??? You can see some of my thoughts in the post. I even generated a Taylor series for the sexp(z) for that base, as well as plots in the complex plane showing some of the misbehavior as real(z) gets larger or smaller. The misbehavior is due to the rationally indifferent fixed point.

One thing I see is that halfway around eta, for real bases between 1 and eta, both fixed points are real, which isn't vertically aligned. For the merged fixed point solution I posted, there would be singularities for the slog/Fatou-coordinate for both fixed points at the real axis. But for bases less than halfway around the circle, then the slog/Fatou-coordinate appears be defined from -infinity to infinity, for the examples I tried.

- Sheldon

(03/07/2012, 12:08 AM)sheldon Wrote: http://math.eretrandre.org/tetrationforu...25#pid6325

I am fascinated by the concept of merged solutions on the Shell Thron boundary itself... Also, Dimitrii reports that his method works well on the Shell Thron boundary....