03/06/2013, 11:57 PM
(This post was last modified: 03/07/2013, 12:23 AM by sheldonison.)
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 \( L=\frac{1}{e} \) where \( b^L=L \) and
\( b={(\frac{1}{e})}^e\approx 0.065988 \)
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.
\( b^{L+x}\approx L-x+1.35914x^2-1.23151x^3+... \)
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 Leau-Fatou 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.
\( b^{b^{L+x}}\approx L+x-1.2315x^3+0.836897x^4+... \)
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 \( \exp(\frac{2\pi i}{5}) \), 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/tetrationforu...hp?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 \( \Im(z) \) 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
\( b={(\frac{1}{e})}^e\approx 0.065988 \)
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.
\( b^{L+x}\approx L-x+1.35914x^2-1.23151x^3+... \)
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 Leau-Fatou 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.
\( b^{b^{L+x}}\approx L+x-1.2315x^3+0.836897x^4+... \)
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 \( \exp(\frac{2\pi i}{5}) \), 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/tetrationforu...hp?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 \( \Im(z) \) 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
