03/07/2012, 12:08 AM
(This post was last modified: 02/21/2013, 08:31 AM by sheldonison.)

I am fascinated by the concept of merged solutions on the Shell Thron boundary itself. My program is now working reasonably well, and my program will attempt numerical results fairly close to the Shell Thron boundary, but the numerical method my algorithm uses cannot work on the boundary itself, where the period for one of the fixed points is real. Initially, I would have thought that real tetration could not be extended analytically past the Shell Thron boundary, but I no longer feel this is the case due to the numerical results I'm seeing, at least for the first crossing of the Shell Thron boundary. Also, Dimitrii reports that his method works well on the Shell Thron boundary. I am also going to post my own novel results for bases on the boundary below, that were developed by looking at analytic merged tetration solutions in a region of points.

But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.

But this post is about the first Shell Thron boundary crossing. At the boundary, as long as the period is irrational (there are some additional restrictions), then the Siegel disc can be wrapped around the unit circle. And the unit circle can be unwrapped, so that the upper superfunction can be defined in the complex plane. But, the edge of the siegel disc is a fractal whose properties depend on the continued fraction of the period. So, briefly, let us briefly review the merged sexp solution generated from the two fixed points.

edit Feb 21st 2013: here, the term SuperFunction refers to the inverse abel function generated from the fixed point via the classic Schroeder function equation. , where S is the Schroeder function and is the fixed point for that Schroeder function, and is the inverse of the classic abel function generated from the Schroeder function. end edit.

if imag(z)>0

if imag(z)<0

is a 1-cyclic function which decays to a constant as z goes to +imaginary infinity, and is also a 1-cyclic function, which decays to a constant as z goes towards -imaginary infinity. At the real axis, we have the merger of the two functions, with sexp(-1)=0, and sexp(0)=1, and sexp(1)=base, and with singularities at integer values of z<=-2. theta(z) for repelling superfunctions has singularities at integer values at the real axis. theta(z) for an attracting superfunction also has singularities, but the singularities need not be at the real axis since the attracting superfunction includes the sexp(z)=0 case. This is not pertinent to my main point though. What is pertinent, is that if a merged at the Shell Thron boundary behaves analytically in a region around the z=0 point, than that implies that the function has to exactly cancel the fractal singularities of the siegel disc boundary, in the neighborhood of z=0.

This might be reason enough to consider whether such solutions exist, but that is not all! If the period is rational, then the function cannot be normalized around a siegel disc, and as far as I know, conventional methods would not be able to produce an analytic superfunction from the rationally indifferent fixed point. This means for a rational period, there is no upper superfunction, and therefore, it would seem that there cannot be a merged fixed point solution at the Shell Thron boundary. Both of these taken together should lead one to doubt whether analytic tetration can be smoothly extended through the Shell Thron boundary. The combination of these facts makes me very interested in the nature of solutions at the Shell Thron boundary.

to be continued, with graphs, and surprising numerical results ....

But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.

But this post is about the first Shell Thron boundary crossing. At the boundary, as long as the period is irrational (there are some additional restrictions), then the Siegel disc can be wrapped around the unit circle. And the unit circle can be unwrapped, so that the upper superfunction can be defined in the complex plane. But, the edge of the siegel disc is a fractal whose properties depend on the continued fraction of the period. So, briefly, let us briefly review the merged sexp solution generated from the two fixed points.

edit Feb 21st 2013: here, the term SuperFunction refers to the inverse abel function generated from the fixed point via the classic Schroeder function equation. , where S is the Schroeder function and is the fixed point for that Schroeder function, and is the inverse of the classic abel function generated from the Schroeder function. end edit.

if imag(z)>0

if imag(z)<0

is a 1-cyclic function which decays to a constant as z goes to +imaginary infinity, and is also a 1-cyclic function, which decays to a constant as z goes towards -imaginary infinity. At the real axis, we have the merger of the two functions, with sexp(-1)=0, and sexp(0)=1, and sexp(1)=base, and with singularities at integer values of z<=-2. theta(z) for repelling superfunctions has singularities at integer values at the real axis. theta(z) for an attracting superfunction also has singularities, but the singularities need not be at the real axis since the attracting superfunction includes the sexp(z)=0 case. This is not pertinent to my main point though. What is pertinent, is that if a merged at the Shell Thron boundary behaves analytically in a region around the z=0 point, than that implies that the function has to exactly cancel the fractal singularities of the siegel disc boundary, in the neighborhood of z=0.

This might be reason enough to consider whether such solutions exist, but that is not all! If the period is rational, then the function cannot be normalized around a siegel disc, and as far as I know, conventional methods would not be able to produce an analytic superfunction from the rationally indifferent fixed point. This means for a rational period, there is no upper superfunction, and therefore, it would seem that there cannot be a merged fixed point solution at the Shell Thron boundary. Both of these taken together should lead one to doubt whether analytic tetration can be smoothly extended through the Shell Thron boundary. The combination of these facts makes me very interested in the nature of solutions at the Shell Thron boundary.

to be continued, with graphs, and surprising numerical results ....