# Tetration Forum

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(04/17/2012, 12:48 PM)sheldonison Wrote: [ -> ]edited Where both fixed points are repelling, there is only one solution, and the fixed points cannot be swapped (check your math, and/or else post of sexp(z)). Henryk has a paper with a uniquess criteria for sexp(z), and I hope to eventually show the theta(z) uniqueness criteria I posted yesterday, $\theta(z)=\text{slog}(f(z))-z$, can be shown to be complete if both fixed points are repelling and theta(z) has a singularity.

So I guess I was right -- the strange fixpoint-reversal phenomenon probably simply means the continuation is impossible.

(04/17/2012, 12:48 PM)sheldonison Wrote: [ -> ]But, inside the Shell Thron boundary, where one of the fixed points is attracting, there are two different solutions for the same base, which also makes the uniqueness criteria more complicated. Are you calculating sexp(z), or slog(z)? I've started to make plots of slog(z) as well, for the bipolar solutions, inside the Shell Thron boundary. Also, as you point out, as you approach the second Shell Thron crossing, the singularities for sexp(z) in the upper half of the complex plane (rotating around eta counterclockwise) start to bunch up and get arbitrarily close to each other and the real axis.
- Sheldon

I was calculating the fixed points to see how the sickel would behave at the boundary crossing, to see if it "exploded" or something weird like that.

How do you graph slog(z), anyway -- i.e. how do you handle all the branch cutting and so forth?
(03/20/2010, 08:12 AM)mike3 Wrote: [ -> ]However, for it to be not real valued in $(1, e^{1/e}]$ and real valued in $(e^{1/e}, \infty)$ would imply there is a singularity/branchpoint at $b = e^{1/e}$, hence not holomorphic there, eh?

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 $\mathcal{F}_1$, 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 $\mathcal{F}_1$. And inside the STR I guess the fixpoints are rather horizontally aligned which also means its not in $\mathcal{F}_1$.

(Of course I always mean here the corresponding meaning of $\mathcal{F_1}$ if the first fixpoint is not set to 0.)

On the other hand if I remember, Sheldon posted somewhere that he also found a solution for the non-primary fixpoints. The secondary fixpoints however do not collapse into a horizontal fixpoint pair, but they remain vertical when passing $b=e^{1/e}$ on the real axis. Though I really wonder whether there can exist a sickle between these fixpoints, i.e. an area bounded by an (injective) curve between both fixpoints and its image under $b^z$.
(02/19/2013, 11:22 PM)bo198214 Wrote: [ -> ]On the other hand if I remember, Sheldon posted somewhere that he also found a solution for the non-primary fixpoints. The secondary fixpoints however do not collapse into a horizontal fixpoint pair, but they remain vertical when passing $b=e^{1/e}$ on the real axis. Though I really wonder whether there can exist a sickle between these fixpoints, i.e. an area bounded by an (injective) curve between both fixpoints and its image under $b^z$.

http://math.eretrandre.org/tetrationforu...hp?tid=452
There's a lot of good posts here, but one quick comment on the non-primary fixed point solution which I posted. The solution from the alternative fixed point doesn't have an analytic abel function at real axis! That's because the first and second derivatives of that sexp(z) goes to zero at every integer>=-2, which leads to singularities for the abel function for sexp(n) where the derivative gets infinitely large at the real axis. So theorems about the Fatou coordinate (or Abel function) wouldn't apply to this weird alternative fixed point solution. Only the solution from the primary fixed point has an analytic abel function at the real axis, with the derivative of the sexp(z) function>0 for all z>-2.
- Sheldon
(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 $\mathcal{F}_1$, 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 $\mathcal{F}_1$. And inside the STR I guess the fixpoints are rather horizontally aligned which also means its not in $\mathcal{F}_1$.

(Of course I always mean here the corresponding meaning of $\mathcal{F_1}$ 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 $b\approx1.96514 + 0.441243i$, 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....

(02/21/2013, 09:01 AM)sheldonison Wrote: [ -> ]We did see some results for bases both inside and outside the Shell Thron boundary.

Oh Sheldon, I am really too slow to catch up with all your findings. Indeed now that I plotted some sickles it seems to me the Shell/Thron boundary is not much of an osbtacle anymore. Whether this follows from the propositions of Shishikura is perhaps another matter.

So the general assumption in the moment is that we can continue tetration along bases through the STB?
What about $\eta$ itself, do we conjecture holomorphy there?

What happened anyway with writing your article. I really find that its a novel efficient way to compute tetration, worth that the world should know it!
(02/24/2013, 08:57 PM)bo198214 Wrote: [ -> ]
(02/21/2013, 09:01 AM)sheldonison Wrote: [ -> ]We did see some results for bases both inside and outside the Shell Thron boundary.

Oh Sheldon, I am really too slow to catch up with all your findings. Indeed now that I plotted some sickles it seems to me the Shell/Thron boundary is not much of an osbtacle anymore. Whether this follows from the propositions of Shishikura is perhaps another matter.

So the general assumption in the moment is that we can continue tetration along bases through the STB?
I have gotten results on the STB itself; although the calculation is indirect, and requires calculating complex tetration for many dozens of complex bases in a circle. See http://math.eretrandre.org/tetrationforu...e=threaded for the irrationally indifferent case and the next entry in that thread for the rationally indifferent case. I think the rationally indifferent case, with a "pseudo" period=5, base=1.96514 + 0.441243i, is a very interesting case, and I was hoping someone had calculated the analogous problem for Mandelbrots inverse perturbed-fatou coordinates on the main cardioid boundary, with a rational Pseudo period.

Dimitrii's also reports that his Cauchy integral algorithm can calculate bases on the Shell Thron boundary directly. I have also extended my indirect results to z anywhere in the complex plane for bases on the STB.

Quote:What about $\eta$ itself, do we conjecture holomorphy there?

What happened anyway with writing your article. I really find that its a novel efficient way to compute tetration, worth that the world should know it!
$\eta$ has a branch singularity, since starting with a real base $>\eta$ the result going halfway around the circle clockwise to a real base<eta is different than the result going halfway around the circle counterclockwise. The tetration function is not real valued. Numerically, the branch singularity at eta appears to be very slight, even at $\eta-0.25$. See http://math.eretrandre.org/tetrationforu...e=threaded Continuing on past $\eta$ more than halfway around the circle starting from a real base $>\eta$, it seems that one encounters a singularity wall when one gets to the STB boundary the second time. Douady saw the same phenomena for $z^2+\frac{1}{4}+\epsilon$. Problem 6) (Douady) Persistence of the Fatou Coordinate:

I'll try to get to that article ... thanks for the encouragement. I still have much to learn.
- Sheldon
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