sexp for base (1/e)^e ~= 0.0660?  sheldonison  03/06/2013
Can it have a Tetration superfunction with sexp(1)=0, sexp(0)=1, sexp(1)=b, sexp(2)=b^b? This is a base on the shellthron boundary, with one fixed point where and
This base has a rationally indifferent multiplier=1, which has a pseudo period=2. Because the multiplier is an exact root of unity, rather than an irrational root of unity, this base doesn't have a Schroeder function.
Iterating this function starting with z=1 doesn't lead to behavior that seem much like Tetration. I was curious if it was possible to generate an sexp(z) solution given that I think there is only one fixed point, and that the fixed is rationally indifferent, in this case with a pseudo period of 2.
As I understand it, the abel function for this function, would be generated from b^b^z. If you iterate the b^z function twice, b^b^z would be a parabolic case with an Abel function for each of the four leaves on the LeauFatou flower. But I don't know how you can combine these four Abel functions together into a single inverse abel function for b^z. Note in the equation below, that there is no x^2 term.
At the real axis starting close to the fixed point and iterating twice always gets you a little bit closer to the fixed point, but the convergence is very slow. For example, starting with z=0, iterating z=b^z oscillates towards the fixed point.
I generated some results and graphs for a complex base on the shellthron boundary with a rationally indifferent fixed point with a multiplier of , with a pseudo period=5, which also lacks a Schroeder function because the multiplier is a 5th root of unity; the other fixed point is repelling. See post #10 of this thread, http://math.eretrandre.org/tetrationforum/showthread.php?tid=729 This result was generated via the conjectured merged complex tetration solution, using both fixed points of the base. It seems that if there is a solution for other rationally indifferent roots of unity on the Shell Thron boundary, than there could be a solution for this base as well.
In the upper half of the complex plane, such a solution for base (1/e)^e might only be well behaved near the imaginary axis, exponentially converging towards the fixed point as increases much like the conjectured solution with pseudo period=5. Such a solution would become increasingly chaotic as real(z) increased or decreased. I have no idea what the solution could look like in the lower half of the complex plane, since I think there is only one fixed point.
 Sheldon
RE: sexp for base (1/e)^e ~= 0.0660?  tommy1729  03/07/2013
I believe in the merged conjecture.
tommy1729
RE: sexp for base (1/e)^e ~= 0.0660?  mike3  03/08/2013
(03/06/2013, 11:57 PM)sheldonison Wrote: Can it have a Tetration superfunction with sexp(1)=0, sexp(0)=1, sexp(1)=b, sexp(2)=b^b? This is a base on the shellthron boundary, with one fixed point where and
This base has a rationally indifferent multiplier=1, which has a pseudo period=2. Because the multiplier is an exact root of unity, rather than an irrational root of unity, this base doesn't have a Schroeder function.
Iterating this function starting with z=1 doesn't lead to behavior that seem much like Tetration. I was curious if it was possible to generate an sexp(z) solution given that I think there is only one fixed point, and that the fixed is rationally indifferent, in this case with a pseudo period of 2.
As I understand it, the abel function for this function, would be generated from b^b^z. If you iterate the b^z function twice, b^b^z would be a parabolic case with an Abel function for each of the four leaves on the LeauFatou flower. But I don't know how you can combine these four Abel functions together into a single inverse abel function for b^z. Note in the equation below, that there is no x^2 term.
At the real axis starting close to the fixed point and iterating twice always gets you a little bit closer to the fixed point, but the convergence is very slow. For example, starting with z=0, iterating z=b^z oscillates towards the fixed point.
I generated some results and graphs for a complex base on the shellthron boundary with a rationally indifferent fixed point with a multiplier of , with a pseudo period=5, which also lacks a Schroeder function because the multiplier is a 5th root of unity; the other fixed point is repelling. See post #10 of this thread, http://math.eretrandre.org/tetrationforum/showthread.php?tid=729 This result was generated via the conjectured merged complex tetration solution, using both fixed points of the base. It seems that if there is a solution for other rationally indifferent roots of unity on the Shell Thron boundary, than there could be a solution for this base as well.
In the upper half of the complex plane, such a solution for base (1/e)^e might only be well behaved near the imaginary axis, exponentially converging towards the fixed point as increases much like the conjectured solution with pseudo period=5. Such a solution would become increasingly chaotic as real(z) increased or decreased. I have no idea what the solution could look like in the lower half of the complex plane, since I think there is only one fixed point.
 Sheldon
Hmm. We can track the two "principal" fixed points in the plane as they go from, say, base 2, to this base via a nonreal contour from above (since I think lies on the branch cut from , and we "conventionally" define such functions to be "continuous from above" (if the cut goes to the right, then it is "continuous from below").). The upper fixed point should be and the lower , assuming my method of repeated Newton method + slowly inching the base toward through a complex plane contour worked. is neutral, as you already know, but is not, and we should be able to generate the regular superfunction there.
The interesting part is that is not a fixed point on the principal branch of the base logarithm. Instead it is fixed on the branch with , i.e. , where is the principal branch of the natural log.
The superfunction generated from the "lower" fixed point looks like this:
[attachment=1005]
(scale is from 10 to +10 on both axes).
So I'd guess that it should look like that in the lower halfplane.
Not sure what to do about the upper halfplane, though.
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  03/08/2013
(03/08/2013, 06:18 AM)mike3 Wrote: Hmm. We can track the two "principal" fixed points in the plane as they go from, say, base 2, to this base via a nonreal contour from above (since I think lies on the branch cut from , and we "conventionally" define such functions to be "continuous from above" (if the cut goes to the right, then it is "continuous from below").). The upper fixed point should be and the lower ....
The interesting part is that is not a fixed point on the principal branch of the base logarithm. Instead it is fixed on the branch with , i.e. , where is the principal branch of the natural log.
The superfunction generated from the "lower" fixed point looks like this:
(scale is from 10 to +10 on both axes).
So I'd guess that it should look like that in the lower halfplane.
Not sure what to do about the upper halfplane, though. Hey Mike, nice graph, thanks! It definitely looks like a superexponentially growing sexp function! I had figured out the repelling fixed points sometime yesterday afternoon, and had generated from the repelling fixed point you gave. Then I figured out the lower superfunction would have a period=~2.04784+2.1555i, which visually matches your plot  that's as far as I got. In between going to infinity, your superfunction visually seems to approach 1/e in a similar way as it would in the upper half of the complex plane for the conjectured merged sexp(z), so that seems where the sexp(z) would stitch together with the superfunction at the real axis. From above with Im(z)>0, the conjectured sexp(z) would approach a two periodic approximation winding around 1/e, .
I'm not ready to compute such an sexp(z) function; the merged sexp(z) algorithm for complex bases doesn't work directly for bases on the Shell Thron boundary for rationally indifferent cases, because such bases lack a Schroeder function from the upper fixed point. The algorithm could work for irrationally indifferent bases on the Shell Thron boundary, even those arbitrarily close to his base. Anyway, one can conjecture this sexp(z) exists. Now, one can imagine following sexp(z) on the Shell Thron boundary counterclockwise from eta to this base, which has a Pseudo period=2. What happens if we continue all the way around back to eta? Does the Pseudo period approach 1 in the upper half of the complex plane, winding around e, approaching a 1cyclic approximation of ? In the lower half of the complex plane would this it approach a different nonprimary fixed point of eta? I also suspect/hope that there is some sort of a generic form for expressing superfunctions for rationally indifferent fixed points superfunctions. I'll post more details later. At the very last, I'll post a more accurate approximation than , for the superfunction for this tetration base in the upper half of the complex plane.
 Sheldon
RE: sexp for base (1/e)^e ~= 0.0660?  mike3  03/09/2013
(03/08/2013, 02:30 PM)sheldonison Wrote: (03/08/2013, 06:18 AM)mike3 Wrote: Hmm. We can track the two "principal" fixed points in the plane as they go from, say, base 2, to this base via a nonreal contour from above (since I think lies on the branch cut from , and we "conventionally" define such functions to be "continuous from above" (if the cut goes to the right, then it is "continuous from below").). The upper fixed point should be and the lower ....
The interesting part is that is not a fixed point on the principal branch of the base logarithm. Instead it is fixed on the branch with , i.e. , where is the principal branch of the natural log.
The superfunction generated from the "lower" fixed point looks like this:
(scale is from 10 to +10 on both axes).
So I'd guess that it should look like that in the lower halfplane.
Not sure what to do about the upper halfplane, though. Hey Mike, nice graph, thanks! It definitely looks like a superexponentially growing sexp function! I had figured out the repelling fixed points sometime yesterday afternoon, and had generated from the repelling fixed point you gave. Then I figured out the lower superfunction would have a period=~2.04784+2.1555i, which visually matches your plot  that's as far as I got. In between going to infinity, your superfunction visually seems to approach 1/e in a similar way as it would in the upper half of the complex plane for the conjectured merged sexp(z), so that seems where the sexp(z) would stitch together with the superfunction at the real axis. From above with Im(z)>0, the conjectured sexp(z) would approach a two periodic approximation winding around 1/e, .
I'm not ready to compute such an sexp(z) function; the merged sexp(z) algorithm for complex bases doesn't work directly for bases on the Shell Thron boundary for rationally indifferent cases, because such bases lack a Schroeder function from the upper fixed point. The algorithm could work for irrationally indifferent bases on the Shell Thron boundary, even those arbitrarily close to his base. Anyway, one can conjecture this sexp(z) exists. Now, one can imagine following sexp(z) on the Shell Thron boundary counterclockwise from eta to this base, which has a Pseudo period=2. What happens if we continue all the way around back to eta? Does the Pseudo period approach 1 in the upper half of the complex plane, winding around e, approaching a 1cyclic approximation of ? In the lower half of the complex plane would this it approach a different nonprimary fixed point of eta? I also suspect/hope that there is some sort of a generic form for expressing superfunctions for rationally indifferent fixed points superfunctions. I'll post more details later. At the very last, I'll post a more accurate approximation than , for the superfunction for this tetration base in the upper half of the complex plane.
 Sheldon
However, I don't think the upperplane will be periodic, since it appears that applying repeatedly to close to seems to lead to a slow and oscillatory attraction toward . I suspect for all .
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  03/09/2013
Quote:However, I don't think the upperplane will be periodic, since it appears that applying repeatedly to close to seems to lead to a slow and oscillatory attraction toward . I suspect for all .
I'm not looking for an exact 2periodic solution; but the solution I'm looking for has limiting behavior that looks more and more like a periodic function as imag(z) goes to infinity. I have the example for base=1.96514 + 0.441243i, which is an indifferent with pseudo period=5, whose sexp(z) solution as imag(z) increases to infinity looks increasingly like . Anything else I would post now would be gibberish, because I haven't gotten a form for the solution yet, even though I can almost see it
 Sheldon
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  03/10/2013
(03/09/2013, 08:25 AM)sheldonison Wrote: I'm not looking for an exact 2periodic solution; but the solution I'm looking for has limiting behavior that looks more and more like a periodic function as imag(z) goes to infinity.
 Sheldon
Here is a better three term approximation for the superfunction for base=(1/e)^e. I don't have a closed form. From the upper fixed point, the superfunction of z is approximately as follows; this approximation has smaller and smaller error has imag(z) gets larger.
superfunction(z)
Notice the third coefficient isn't your standard three periodic function, since we multiply the periodic function by z. These coefficients are half the coefficients of b^(1/e + z). The solution isn't exactly 2periodic, but that's because the base function is rationally indifferent with a pseudo period=2. I don't have a closed form, and I think the coefficients get even more complicated for the fourth term and above. The error term for this function is roughly . As you iterate b^z, or its inverse, the function changes with each double iteration, and becomes less well behaved the farther you get from the imaginary axis.
 Sheldon
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  11/13/2013
(03/10/2013, 06:35 AM)sheldonison Wrote: I was inspired by Mike's http://math.eretrandre.org/tetrationforum/showthread.php?tid=828&pid=6742#pid6742 recent post to go back and work this some more. Here is an even better multiple term approximation for the superfunction for base=. I would imagine this might even have the mythical infinite form that would converge exactly, that I imagined must exist, since we see tetration solutions for rationally indifferent complex bases.....
superfunction(z)
I don't know how to find the equivalent function out on the web. The coefficients are real valued linear combinations of the coeffients for B^L. They were generated in the order they were listed, by iteratively solving the equation requiring that B^superfunction(z)=superfunction(z+1), for each partial term, one at a time. The algebra is messy, but I finally got it the algebra to work. Note that for odd terms, the periodic term must be multiplied by z since . So all of the coefficients from n=3 upward are much more complex "multiple" term coefficients, multplied by z. Starting with n=5 and n=6, we also need both a z^2 multiplier, and a "z" multiplier. At , each new "multiple term" increases the accuracy by approximately 23x, and at , each new "multiple term" increases accuracy by 500x, so that the equation above at 2i, is accurate to , and at i, it is accurate to
Even with the formal solution, this equation quickly misbehaves, iterating in either direction, as you increase or decrease the real part of z, but this is also expected, due to the crazy behavior of rationally indifferent fixed points. The more amazing part is that this crazy behavior can be stitched together with a theta mapping (we think), with the other fixed point, and lead to something well behaved in the negative imaginary part of the complex plane too!
I have also generated a similar solution for iterating , which is also a rationally indifferent 2periodic point in the Mandelbrot set, whose coefficients turn out to be simpler, and rational valued. For this mapping, the fixed point is 1/2. I've written a parigp program that can automatically generate these multiple term coefficients, presumably for arbitrarily large n, though I've only generated them for these two cases. Again, the existence of this type of solution has been puzzling me for a long time, and I would really like to see someone else's work on a similar problem.
Iterating: ,
superfunction(z)
 Sheldon
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  11/13/2013
(03/08/2013, 06:18 AM)mike3 Wrote: ....
The superfunction generated from the "lower" fixed point looks like this:
(scale is from 10 to +10 on both axes).
So I'd guess that it should look like that in the lower halfplane.
Not sure what to do about the upper halfplane, though.
Ok, here's a shot at the upper half of the complex plane, based on what I computed yesterday, generated from an 11 "multiterm" approximation, . It is pretty clear that one could probably stitch these two functions together, this upper function, and the lower function you generated from the L=0.196+1.691i fixed point; by choosing the logarithmic branch to be to the left of the first singularity, seen near 1.99+0.23i. From the left of the singularity, there is a common path between the two functions. The scale here is 2x the scale of your plot, from 5 to +5 on the real axis, and from 0 to 2 on the imaginary axis.
 Sheldon
[attachment=1033]
EDIT: Here is the upper fixed point function from 4 to +2, at 0.15i which is just under the logarithmic singularity branch, where the singularity at approx 1.99+0.23i. If the logarithm branch is chosen correctly, there is a very clear path to the left, going to the secondary fixed point, of 0.196+1.691i.
 Sheldon
[attachment=1032]
RE: sexp for base (1/e)^e ~= 0.0660?  sheldonison  11/22/2013
Here is the merged tetration solution for . The graph extends from 4.12 to 6.12 in the real direction, and 2.12 to 6.12 in the imaginary direction, with grid lines every 2 units. At the origin, sexp(0)=1. You can see the branch singularity at sexp(2), where from below the function quickly converges to the regular iteration from the fixed point of 0.1957+1.691i, and from above, as increases, the merged tetration function converges to and to the novel pseudo 2periodic solution I came up with earlier in this post. Near the origin, at z=0, you can see looking at the graph, that the function misbehaves as imag(z) decreases, before it starts behaving again. On the unit circle centered where sexp(0)=1, the function has a maximum magnitude near x=0.60.8i, of around 35000! This behavior makes it difficult to find an initial seed for the sexp(z) solution, or for Kouznetsov's method. I used a 10term version of the "multiterm" exponential series that I posted earlier, as a starting seed for the upper half of the complex plane. The convergence was good enough to generate the lower half plane theta mapping from the other fixed point, using the path that is in my previous post. From there, I generated an initial sexp(z) taylor series on a unit circle, and then I iterated generating an upper half plane theta mapping and a lower half plane theta mapping, and an sexp Taylor series. I used the regular superfunction from the other fixed point, along with the 10term "multiterm" pseudo 2cyclic series I generated, from the exp(e) fixed point. The sexp Taylor series, posted below the picture, appears to be accurate to more than 10 decimal digits.
[attachment=1035]
I realize of course that this "multiterm" series I used for the upper half plane theta mapping doesn't have any of the strong theoretical underpinnings of a Schroder series, but it does seem to converge, and I think the results are at least interesting. This merged tetration solution has other singularities in the upper half of the complex plane, for real(z)<2. For example, the second singularity is near z=4.006+0.076i, and is visible near the left edge of the graph. Perhaps Mike can round off this sexp(z) as a starting seed for his new method. Also, Mike reports getting results for b=0.04, where both fixed points are repelling. Was that tetration solution as poorly behaving as the results for b=exp(e)?
Code: taylor series, for sexp(z=0), sexp(0)=1, for base exp(e). sexp(z1) is a little better behaved; posted below.
{sexp=
1.000000000000000000 +
+x^ 1* ( 0.2139536657103389784 + 2.528819288258810236*I)
+x^ 2* (5.718005651873086168 + 1.405529567012070011*I)
+x^ 3* (4.994351362455816907  9.591927497023705884*I)
+x^ 4* ( 12.47110195303975205  12.56438589624159071*I)
+x^ 5* ( 24.64512057624859248 + 11.06869551290638396*I)
+x^ 6* (1.004557358147892928 + 39.11086617384984950*I)
+x^ 7* (50.04117943660604027 + 21.13926212969639756*I)
+x^ 8* (54.67981005276769267  48.16965142184401020*I)
+x^ 9* ( 23.82775802285444627  91.92324075543739587*I)
+x^10* ( 117.4806865647258164  27.71925893384273284*I)
+x^11* ( 100.3038670585419754 + 111.7102325184134650*I)
+x^12* (58.72453764469395163 + 173.5093912929210437*I)
+x^13* (215.3969244783325557 + 43.18987024699570534*I)
+x^14* (173.4573393798765433  192.8890001580594647*I)
+x^15* ( 87.68637672585892981  289.0206055188629291*I)
+x^16* ( 335.6745790101675781  88.48727597802993670*I)
+x^17* ( 287.6967374498740693 + 269.6080191033535142*I)
+x^18* (81.76939211911840563 + 435.3167743926416319*I)
+x^19* (455.8704451399854484 + 186.4573068027563207*I)
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+x^96* (5.992699854901776532 + 4.190319687722143593*I)
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