Well, for the case it is needed, here is some more explanation (see a more general remark at the end).
In the "regular tetration" (this is that method, where we use the exponential series wich is recentered around a fixpoint, say the lower (attracting) fixpoint \( t_0 \) ) we realize the tetration to fractional heights via the "Schröder"-function (see wikipedia), say
\( \hspace{48} s_0= S (x - t_0) \) where \( S(\cdot) \) denotes the Schröder function.
After that we calculate the "height"-parameter, say "h" into it, where "h" goes into the exponent of the log of the fixpoint:
\( \hspace{48} r_0 = s_0 \cdot \log(t_0)^h \)
Then we use the inverse Schröder-function to find the value \( x_h \) which is the (fractional) h'th iterate "from" \( x_0 \) :
\( \hspace{48} x_h = S^{\circ -1} (r_0) + t_0 \)
Now if we let the "height" h equal zero, thus no iteration, but just change the sign of \( r_0 = -s_0 \) then we get a "dual" \( \tilde x_0 \) , where if \( x_0 \) is between 2 and 4, then the dual is below 2, and if \( x_0 \) is below 2 then its "dual" is between 2 and 4.
This is what I meant with "dual" or "a pair of related numbers".
This simple idea reduces to the formula:
\( \hspace{48} \tilde x_0 = S^{\circ -1} (-s_0) + t_0 = S^{\circ -1} ( - S(x_0 - t_0)) + t_0 \)
But the effect of changing sign in \( s_0 \) is alternatively reachable, if we simply introduce an imaginary value for the height-parameter, since \( \hspace{48} -s_0 = s_0 \cdot \ln(t_0) ^{i \pi \over \ln \ln t_0} \)
Now, starting at some \( x_0 \) between 2 and 4 we can infinitely iterate and at most approach 2, but we can never arrive with any real height a value below 2 by any number of iterations. We might say, that 2 is the infinite iteration from that \( x_0 \).
We see by this that this concept of "imaginary height" (an imaginary value in the height-parameter h) allows to proceed not only to the value 2 but to values below 2. Thus I said with a sloppy expression: "imaginary height can overstep infinite height".
Now, if we take the dual of, say \( x_0 = 0 \) this gives something above 2. If we iterate this one time towards 4 (which means with one negative height), we get that mentioned value of 2.764... . And its dual is then negative infinity (which itself is \( x_0=0 \) iterated one time with negative height - a thing which is not possible otherwise because of the occuring singularity).
Remark: just to restate it again: "my matrix-method" is nothing else than the regular tetration. I came to this by the (accidentally) rediscovery of the concept of Carleman-matrices (see also wikipedia) not knowing that name, and where I was always working under the assumption of infinite size (no truncation), which has some sophisticated impact against a concept of truncated matrices for instance for the conception of fractional powers
In the "regular tetration" (this is that method, where we use the exponential series wich is recentered around a fixpoint, say the lower (attracting) fixpoint \( t_0 \) ) we realize the tetration to fractional heights via the "Schröder"-function (see wikipedia), say
\( \hspace{48} s_0= S (x - t_0) \) where \( S(\cdot) \) denotes the Schröder function.
After that we calculate the "height"-parameter, say "h" into it, where "h" goes into the exponent of the log of the fixpoint:
\( \hspace{48} r_0 = s_0 \cdot \log(t_0)^h \)
Then we use the inverse Schröder-function to find the value \( x_h \) which is the (fractional) h'th iterate "from" \( x_0 \) :
\( \hspace{48} x_h = S^{\circ -1} (r_0) + t_0 \)
Now if we let the "height" h equal zero, thus no iteration, but just change the sign of \( r_0 = -s_0 \) then we get a "dual" \( \tilde x_0 \) , where if \( x_0 \) is between 2 and 4, then the dual is below 2, and if \( x_0 \) is below 2 then its "dual" is between 2 and 4.
This is what I meant with "dual" or "a pair of related numbers".
This simple idea reduces to the formula:
\( \hspace{48} \tilde x_0 = S^{\circ -1} (-s_0) + t_0 = S^{\circ -1} ( - S(x_0 - t_0)) + t_0 \)
But the effect of changing sign in \( s_0 \) is alternatively reachable, if we simply introduce an imaginary value for the height-parameter, since \( \hspace{48} -s_0 = s_0 \cdot \ln(t_0) ^{i \pi \over \ln \ln t_0} \)
Now, starting at some \( x_0 \) between 2 and 4 we can infinitely iterate and at most approach 2, but we can never arrive with any real height a value below 2 by any number of iterations. We might say, that 2 is the infinite iteration from that \( x_0 \).
We see by this that this concept of "imaginary height" (an imaginary value in the height-parameter h) allows to proceed not only to the value 2 but to values below 2. Thus I said with a sloppy expression: "imaginary height can overstep infinite height".
Now, if we take the dual of, say \( x_0 = 0 \) this gives something above 2. If we iterate this one time towards 4 (which means with one negative height), we get that mentioned value of 2.764... . And its dual is then negative infinity (which itself is \( x_0=0 \) iterated one time with negative height - a thing which is not possible otherwise because of the occuring singularity).
Remark: just to restate it again: "my matrix-method" is nothing else than the regular tetration. I came to this by the (accidentally) rediscovery of the concept of Carleman-matrices (see also wikipedia) not knowing that name, and where I was always working under the assumption of infinite size (no truncation), which has some sophisticated impact against a concept of truncated matrices for instance for the conception of fractional powers
Gottfried Helms, Kassel
