In the regular tetration we use the Schröder-function for the linearization of the fractional heights-computation beginning at some x0.
Experimenting with it, say tetration with the base \( b=\sqrt 2 \) we have a lower fixpoint \( t_0=2 \). Here the values of the Schröder-function for \( x \gt t_0 \) are positive and that of \( x \lt t_0 \) are negative. Since the effect of changing sign of the Schröder-value is the same as using an imaginary component to the height-parameter, we can say, that iterations from the region above the fixpoint \( t_0 \) down to that below that fixpoint can be achieved by an imaginary height - so, in some sense, the imaginary oversteps the infinite iteration-height.
But this allows to define a pairwise relation between x-values, whose Schröder-values have opposite signs. So \( x=1 \)(below the fixpoint) has the negative schröder-value from \( x=2.46791405... \) (above the fixpoint). Let's call the two related points "duals" of each other.
One can find that the dual of \( x=- \infty \) for base \( b=\sqrt 2 \) is about
\( x_w \approx 2.76432104000012572327981201783... \)
Does someone "know" this value and knows (or with some seriousness guesses) more properties of this value? For instance, what is then the dual of that \( \log_b(x_w) \approx 2.93385035151 \) : is this \( \log_b(-\infty) \) ?
Experimenting with it, say tetration with the base \( b=\sqrt 2 \) we have a lower fixpoint \( t_0=2 \). Here the values of the Schröder-function for \( x \gt t_0 \) are positive and that of \( x \lt t_0 \) are negative. Since the effect of changing sign of the Schröder-value is the same as using an imaginary component to the height-parameter, we can say, that iterations from the region above the fixpoint \( t_0 \) down to that below that fixpoint can be achieved by an imaginary height - so, in some sense, the imaginary oversteps the infinite iteration-height.
But this allows to define a pairwise relation between x-values, whose Schröder-values have opposite signs. So \( x=1 \)(below the fixpoint) has the negative schröder-value from \( x=2.46791405... \) (above the fixpoint). Let's call the two related points "duals" of each other.
One can find that the dual of \( x=- \infty \) for base \( b=\sqrt 2 \) is about
\( x_w \approx 2.76432104000012572327981201783... \)
Does someone "know" this value and knows (or with some seriousness guesses) more properties of this value? For instance, what is then the dual of that \( \log_b(x_w) \approx 2.93385035151 \) : is this \( \log_b(-\infty) \) ?
Gottfried Helms, Kassel
