(08/11/2009, 09:31 PM)jaydfox Wrote:(08/11/2009, 08:05 PM)bo198214 Wrote: So did someone check already whether base conversion of regular iteration gives again regular iteration (say at the lower fixed point)?Well, when I tried a couple years ago, I got different results when using eta and sqrt(2) (using the upper fixed point), so I assume in general that base conversion does not give the same results as regular iteration from the upper fixed point.
It does not even need to be applied on base \( \eta \). Just take two bases \( a,b\in(1,\eta) \) and consider their regular superexponentials at (say) the lower fixed point \( \sigma_a \) and \( \sigma_b \).
Do then both superexponentials transform according to your change of base? I.e. do we have:
\( \sigma_b\circ \sigma^{-1}_a=\lim_{n\to\infty} \log_b^{[n]} \circ\exp_a^{[n]} \).
I doubt, I suppose we see again the wobble.
Quote:I think of it as climbing up a mountain of iterated exponentials in one base, then back down a mountain in the other base (undoing the exponentials by taking logarithms). Just a metaphor, and your mileage may vary.Dont understand me wrong, I like your metaphors, however I also like if you accompany your epic descriptions with some unmistakable formulations and formulas.
